set theory - umd department of computer science · 2016-07-13 · outline 1 branches of set theory...
TRANSCRIPT
![Page 1: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/1.jpg)
Set Theory
Jason Filippou
CMSC250 @ UMCP
06-20-2016
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 1 / 56
![Page 2: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/2.jpg)
Outline
1 Branches of Set Theory
2 Basic DefinitionsSingle setsTwo or more sets
3 Proofs with sets
4 An application: Formal languages
5 Paradoxes in Set Theory
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 2 / 56
![Page 3: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/3.jpg)
Branches of Set Theory
Branches of Set Theory
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 3 / 56
![Page 4: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/4.jpg)
Branches of Set Theory
Naive
Naive set theory is typically taught even at elementary schoolnowadays.
Only kind of set theory till the 1870s!Consists of applications of Venn Diagrams.
Very intuitive, suitable for graphical applications
Not an ounce of formality.
Cannot be used for...
Formal Proofs!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 4 / 56
![Page 5: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/5.jpg)
Branches of Set Theory
Naive
Naive set theory is typically taught even at elementary schoolnowadays.
Only kind of set theory till the 1870s!Consists of applications of Venn Diagrams.
Very intuitive, suitable for graphical applications
Not an ounce of formality.
Cannot be used for... Formal Proofs!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 4 / 56
![Page 6: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/6.jpg)
Branches of Set Theory
Naive
Based entirely on Venn Diagrams.
Ω
Α ΒC
Figure 1: An example Venn Diagram.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 5 / 56
![Page 7: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/7.jpg)
Branches of Set Theory
Axiomatic (Cantor & Dedekind)
First axiomatization of Set Theory.
Understanding of infinite sets and their cardinality.
Figure 2: Georg Cantor, 1870s Figure 3: Richard Dedekind, 1900s
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 6 / 56
![Page 8: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/8.jpg)
Branches of Set Theory
Famous Result
1874 Cantor paper: “On a Property of the Collection of All RealAlgebraic Numbers”
The set of real numbers is uncountable.
Also:
Power set operation.Cantor’s paradise.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 7 / 56
![Page 9: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/9.jpg)
Branches of Set Theory
Famous Result
1874 Cantor paper: “On a Property of the Collection of All RealAlgebraic Numbers”
The set of real numbers is uncountable.
Also:
Power set operation.Cantor’s paradise.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 7 / 56
![Page 10: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/10.jpg)
Branches of Set Theory
Russel’s Paradox
Consider the following set:
S = x|x /∈ x
Then, does S ∈ S ?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 8 / 56
![Page 11: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/11.jpg)
Branches of Set Theory
So why do we care?
If Axiomatic (Cantorian) Set Theory is “broken”, why do westudy it?
Rough answer: Just because a theory is “broken” (i.e leads tocontradictions) doesn’t mean we shouldn’t study it.
Theories are specialized (more stuff is added to them) in order toavoid contradictions all the time.
Non-Euclidean geometries.Zermello - Fraenkel Set Theory.
Qualitative answer: it gives us background necessary to discuss:1 Limitations of computers as a whole!2 Some fundamental results on countability and uncountability of
infinite sets.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 9 / 56
![Page 12: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/12.jpg)
Branches of Set Theory
Branches of Axiomatic Set Theory
Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).
Kripke-Platek
Von Neumann - Bernays - Godel
Morse-Kelley
Tarski-Grothendieck
. . .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56
![Page 13: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/13.jpg)
Branches of Set Theory
Branches of Axiomatic Set Theory
Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).
Kripke-Platek
Von Neumann - Bernays - Godel
Morse-Kelley
Tarski-Grothendieck
. . .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56
![Page 14: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/14.jpg)
Branches of Set Theory
Branches of Axiomatic Set Theory
Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).
Kripke-Platek
Von Neumann - Bernays - Godel
Morse-Kelley
Tarski-Grothendieck
. . .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56
![Page 15: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/15.jpg)
Branches of Set Theory
Branches of Axiomatic Set Theory
Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).
Kripke-Platek
Von Neumann - Bernays - Godel
Morse-Kelley
Tarski-Grothendieck
. . .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56
![Page 16: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/16.jpg)
Branches of Set Theory
Branches of Axiomatic Set Theory
Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).
Kripke-Platek
Von Neumann - Bernays - Godel
Morse-Kelley
Tarski-Grothendieck
. . .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56
![Page 17: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/17.jpg)
Branches of Set Theory
Branches of Axiomatic Set Theory
Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).
Kripke-Platek
Von Neumann - Bernays - Godel
Morse-Kelley
Tarski-Grothendieck
. . .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56
![Page 18: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/18.jpg)
Basic Definitions
Basic Definitions
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 11 / 56
![Page 19: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/19.jpg)
Basic Definitions Single sets
Single sets
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 12 / 56
![Page 20: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/20.jpg)
Basic Definitions Single sets
Definitions
Definition (Set)
A set is a collection of objects without repetitions.
Definition (Ordered Set)
An ordered set is a pair (S,≤), where S is a set and ≤ is a totalorder.
Total orders: binary relations that are antisymmetric,transitive and total.
Examples: ≤, ⊆, lexicographic ordering.
Definition (Multiset)
A multiset is a collection of objects with repetitions.
We won’t really care about those, or about ordered multisets.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56
![Page 21: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/21.jpg)
Basic Definitions Single sets
Definitions
Definition (Set)
A set is a collection of objects without repetitions.
Definition (Ordered Set)
An ordered set is a pair (S,≤), where S is a set and ≤ is a totalorder.
Total orders: binary relations that are antisymmetric,transitive and total.
Examples: ≤, ⊆, lexicographic ordering.
Definition (Multiset)
A multiset is a collection of objects with repetitions.
We won’t really care about those, or about ordered multisets.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56
![Page 22: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/22.jpg)
Basic Definitions Single sets
Definitions
Definition (Set)
A set is a collection of objects without repetitions.
Definition (Ordered Set)
An ordered set is a pair (S,≤), where S is a set and ≤ is a totalorder.
Total orders: binary relations that are antisymmetric,transitive and total.
Examples: ≤, ⊆, lexicographic ordering.
Definition (Multiset)
A multiset is a collection of objects with repetitions.
We won’t really care about those, or about ordered multisets.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56
![Page 23: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/23.jpg)
Basic Definitions Single sets
Membership
Chief operation on sets: membership (∈).
If Ω is a domain of choice and S is a set, any element e of Ω caneither belong to A (e ∈ A) or not (e /∈ A).
Since the chief operation is membership, and sets have uniqueelements, how would you implement them in computer memory?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 14 / 56
![Page 24: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/24.jpg)
Basic Definitions Single sets
Membership
Chief operation on sets: membership (∈).
If Ω is a domain of choice and S is a set, any element e of Ω caneither belong to A (e ∈ A) or not (e /∈ A).Since the chief operation is membership, and sets have uniqueelements, how would you implement them in computer memory?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 14 / 56
![Page 25: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/25.jpg)
Basic Definitions Single sets
Defining a set
Three ways.
1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .
2 Definition: A = z ∈ Z | z ≥ −23 Agreed upon symbol: N, P, etc4 An operation (union, superset, etc): C = A ∪B, P(0, 1)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56
![Page 26: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/26.jpg)
Basic Definitions Single sets
Defining a set
Three ways.1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .
2 Definition: A = z ∈ Z | z ≥ −23 Agreed upon symbol: N, P, etc4 An operation (union, superset, etc): C = A ∪B, P(0, 1)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56
![Page 27: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/27.jpg)
Basic Definitions Single sets
Defining a set
Three ways.1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .
2 Definition: A = z ∈ Z | z ≥ −2
3 Agreed upon symbol: N, P, etc4 An operation (union, superset, etc): C = A ∪B, P(0, 1)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56
![Page 28: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/28.jpg)
Basic Definitions Single sets
Defining a set
Three ways.1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .
2 Definition: A = z ∈ Z | z ≥ −23 Agreed upon symbol: N, P, etc
4 An operation (union, superset, etc): C = A ∪B, P(0, 1)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56
![Page 29: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/29.jpg)
Basic Definitions Single sets
Defining a set
Three ways.1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .
2 Definition: A = z ∈ Z | z ≥ −23 Agreed upon symbol: N, P, etc4 An operation (union, superset, etc): C = A ∪B, P(0, 1)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56
![Page 30: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/30.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 31: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/31.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 32: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/32.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ?
3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 33: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/33.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 34: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/34.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ?
100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 35: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/35.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 36: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/36.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ?
99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 37: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/37.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 38: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/38.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ?
ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 39: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/39.jpg)
Basic Definitions Single sets
Cardinality
Definition (Cardinality of a set)
Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.
aHold your horses, please.
Corollary
∀A ⊆ Ω, |A| ≥ 0
|−10, 0, 10| = ? 3
|n ∈ N|n < 100| = ? 100
|n ∈ N∗|n < 100| = ? 99
|N| = ? ℵ0 (???)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56
![Page 40: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/40.jpg)
Basic Definitions Single sets
Infinite sets
Certain sets are infinite!
Examples: N, P, (a, b, c) ∈ N+|a2 + b2 = c2Their cardinality cannot be expressed in the same way as that ofa finite set.
To talk about cardinality of infinite sets, we need some functionbackground...Namely, bijections, injections, surjections....
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 17 / 56
![Page 41: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/41.jpg)
Basic Definitions Single sets
The empty set
Definition (Empty set)
There exists a unique set with no elements, denoted ∅ or and calledthe empty set.
Corollary
|∅| = 0
Corollary
∀A ⊆ Ω, ∅ ⊆ A
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 18 / 56
![Page 42: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/42.jpg)
Basic Definitions Single sets
The empty set
Definition (Empty set)
There exists a unique set with no elements, denoted ∅ or and calledthe empty set.
Corollary
|∅| = 0
Corollary
∀A ⊆ Ω, ∅ ⊆ A
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 18 / 56
![Page 43: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/43.jpg)
Basic Definitions Single sets
The empty set
Definition (Empty set)
There exists a unique set with no elements, denoted ∅ or and calledthe empty set.
Corollary
|∅| = 0
Corollary
∀A ⊆ Ω, ∅ ⊆ A
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 18 / 56
![Page 44: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/44.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?
1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 45: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/45.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅
2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 46: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/46.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅2 ∅
3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 47: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/47.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅2 ∅3
4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 48: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/48.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . .
5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 49: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/49.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . .
6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 50: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/50.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . .
7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 51: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/51.jpg)
Basic Definitions Single sets
Some practice
How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56
![Page 52: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/52.jpg)
Basic Definitions Single sets
The powerset
Definition (Powerset)
Let A be a set. The powerset of A, denoted P(A), is the set of allsubsets of A.
Corollary
∀A ⊆ Ω, ∅ ∈ P(A)
Corollary
∀A ⊆ Ω, A ∈ P(A)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 20 / 56
![Page 53: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/53.jpg)
Basic Definitions Single sets
The powerset
Definition (Powerset)
Let A be a set. The powerset of A, denoted P(A), is the set of allsubsets of A.
Corollary
∀A ⊆ Ω, ∅ ∈ P(A)
Corollary
∀A ⊆ Ω, A ∈ P(A)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 20 / 56
![Page 54: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/54.jpg)
Basic Definitions Single sets
The powerset
Definition (Powerset)
Let A be a set. The powerset of A, denoted P(A), is the set of allsubsets of A.
Corollary
∀A ⊆ Ω, ∅ ∈ P(A)
Corollary
∀A ⊆ Ω, A ∈ P(A)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 20 / 56
![Page 55: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/55.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ?
∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 56: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/56.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, c
P(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 57: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/57.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ?
∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 58: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/58.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1
P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 59: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/59.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ?
∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 60: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/60.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . .
P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 61: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/61.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ?
∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 62: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/62.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅
P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 63: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/63.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ?
Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 64: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/64.jpg)
Basic Definitions Single sets
Powerset examples
Examples:
P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56
![Page 65: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/65.jpg)
Basic Definitions Single sets
(Absolute) Set complement
Figure 4: Venn Diagramillustrating the absolutecomplement of a set.
Definition (Set Complement)
Let A be a set in the universaldomain Ω. The absolutecomplement of A, denoted A′, isdefined as the set: x ∈ Ω | x /∈ A.
Corollary (Complement of emptyset)
∅′ = Ω
Corollary (Complement ofuniversal domain)
Ω′ = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 22 / 56
![Page 66: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/66.jpg)
Basic Definitions Single sets
(Absolute) Set complement
Figure 4: Venn Diagramillustrating the absolutecomplement of a set.
Definition (Set Complement)
Let A be a set in the universaldomain Ω. The absolutecomplement of A, denoted A′, isdefined as the set: x ∈ Ω | x /∈ A.
Corollary (Complement of emptyset)
∅′ = Ω
Corollary (Complement ofuniversal domain)
Ω′ = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 22 / 56
![Page 67: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/67.jpg)
Basic Definitions Single sets
Caution: Sets and intervals
This: 0, 1, 2, . . . , 10
Is not the same as this: [0, . . . , 10].
Intervals are sets, but sets are not intervals.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 23 / 56
![Page 68: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/68.jpg)
Basic Definitions Single sets
Caution: Sets and intervals
This: 0, 1, 2, . . . , 10Is not the same as this: [0, . . . , 10].
Intervals are sets, but sets are not intervals.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 23 / 56
![Page 69: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/69.jpg)
Basic Definitions Single sets
Exercises
Provide the following sets in interval or set notation. Assume thatΩ = R
(Z∗−)′ = ?(−1, 1] ∪ −1 = ?[0, 10]′ = ?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 24 / 56
![Page 70: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/70.jpg)
Basic Definitions Two or more sets
Two or more sets
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 25 / 56
![Page 71: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/71.jpg)
Basic Definitions Two or more sets
Subset
Figure 5: A subset of a set
Definition (Subset)
A set B is a subset of set A,denoted A ⊆ B, if and only if∀x ∈ Ω, x ∈ B ⇒ x ∈ A
Examples:
N ⊆ ZP ⊆ Qa ∈ Z
∣∣4|a ⊆ Zeven
Definition (Superset)
A set A is a superset of set B,denoted A ⊇ B, if and only ifB ⊆ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56
![Page 72: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/72.jpg)
Basic Definitions Two or more sets
Subset
Figure 5: A subset of a set
Definition (Subset)
A set B is a subset of set A,denoted A ⊆ B, if and only if∀x ∈ Ω, x ∈ B ⇒ x ∈ A
Examples:
N ⊆ Z
P ⊆ Qa ∈ Z
∣∣4|a ⊆ Zeven
Definition (Superset)
A set A is a superset of set B,denoted A ⊇ B, if and only ifB ⊆ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56
![Page 73: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/73.jpg)
Basic Definitions Two or more sets
Subset
Figure 5: A subset of a set
Definition (Subset)
A set B is a subset of set A,denoted A ⊆ B, if and only if∀x ∈ Ω, x ∈ B ⇒ x ∈ A
Examples:
N ⊆ ZP ⊆ Q
a ∈ Z∣∣4|a ⊆ Zeven
Definition (Superset)
A set A is a superset of set B,denoted A ⊇ B, if and only ifB ⊆ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56
![Page 74: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/74.jpg)
Basic Definitions Two or more sets
Subset
Figure 5: A subset of a set
Definition (Subset)
A set B is a subset of set A,denoted A ⊆ B, if and only if∀x ∈ Ω, x ∈ B ⇒ x ∈ A
Examples:
N ⊆ ZP ⊆ Qa ∈ Z
∣∣4|a ⊆ Zeven
Definition (Superset)
A set A is a superset of set B,denoted A ⊇ B, if and only ifB ⊆ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56
![Page 75: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/75.jpg)
Basic Definitions Two or more sets
Subset
Figure 5: A subset of a set
Definition (Subset)
A set B is a subset of set A,denoted A ⊆ B, if and only if∀x ∈ Ω, x ∈ B ⇒ x ∈ A
Examples:
N ⊆ ZP ⊆ Qa ∈ Z
∣∣4|a ⊆ Zeven
Definition (Superset)
A set A is a superset of set B,denoted A ⊇ B, if and only ifB ⊆ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56
![Page 76: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/76.jpg)
Basic Definitions Two or more sets
Proper subset
Definition (Proper subset)
A set A is a proper subset of set B, denoted A ⊂ B, if it is a subsetof set B and ∃x ∈ B : x /∈ A.
Examples:
Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅
Definition (Proper superset)
A set A is a proper superset of set B, denoted A ⊃ B, if and only ifB ⊂ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56
![Page 77: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/77.jpg)
Basic Definitions Two or more sets
Proper subset
Definition (Proper subset)
A set A is a proper subset of set B, denoted A ⊂ B, if it is a subsetof set B and ∃x ∈ B : x /∈ A.
Examples:
Q ⊂ R
1 ⊂ 0, 1 ⊂ ∅, ∅
Definition (Proper superset)
A set A is a proper superset of set B, denoted A ⊃ B, if and only ifB ⊂ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56
![Page 78: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/78.jpg)
Basic Definitions Two or more sets
Proper subset
Definition (Proper subset)
A set A is a proper subset of set B, denoted A ⊂ B, if it is a subsetof set B and ∃x ∈ B : x /∈ A.
Examples:
Q ⊂ R1 ⊂ 0, 1
⊂ ∅, ∅
Definition (Proper superset)
A set A is a proper superset of set B, denoted A ⊃ B, if and only ifB ⊂ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56
![Page 79: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/79.jpg)
Basic Definitions Two or more sets
Proper subset
Definition (Proper subset)
A set A is a proper subset of set B, denoted A ⊂ B, if it is a subsetof set B and ∃x ∈ B : x /∈ A.
Examples:
Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅
Definition (Proper superset)
A set A is a proper superset of set B, denoted A ⊃ B, if and only ifB ⊂ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56
![Page 80: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/80.jpg)
Basic Definitions Two or more sets
Proper subset
Definition (Proper subset)
A set A is a proper subset of set B, denoted A ⊂ B, if it is a subsetof set B and ∃x ∈ B : x /∈ A.
Examples:
Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅
Definition (Proper superset)
A set A is a proper superset of set B, denoted A ⊃ B, if and only ifB ⊂ A.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56
![Page 81: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/81.jpg)
Basic Definitions Two or more sets
Corollaries of subset definition
Corollary (Any set is a subset of itself)
∀S ⊆ Ω, S ⊆ S
Corollary (Any set is an element of its powerset)
∀S ⊆ Ω, S ∈ P(S)
Corollary
∀A,B such that A ⊂ B, A ⊆ B
Corollary
∀S, ∅ ⊆ S
Can we prove the latter corollary?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56
![Page 82: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/82.jpg)
Basic Definitions Two or more sets
Corollaries of subset definition
Corollary (Any set is a subset of itself)
∀S ⊆ Ω, S ⊆ S
Corollary (Any set is an element of its powerset)
∀S ⊆ Ω, S ∈ P(S)
Corollary
∀A,B such that A ⊂ B, A ⊆ B
Corollary
∀S, ∅ ⊆ S
Can we prove the latter corollary?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56
![Page 83: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/83.jpg)
Basic Definitions Two or more sets
Corollaries of subset definition
Corollary (Any set is a subset of itself)
∀S ⊆ Ω, S ⊆ S
Corollary (Any set is an element of its powerset)
∀S ⊆ Ω, S ∈ P(S)
Corollary
∀A,B such that A ⊂ B, A ⊆ B
Corollary
∀S, ∅ ⊆ S
Can we prove the latter corollary?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56
![Page 84: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/84.jpg)
Basic Definitions Two or more sets
Corollaries of subset definition
Corollary (Any set is a subset of itself)
∀S ⊆ Ω, S ⊆ S
Corollary (Any set is an element of its powerset)
∀S ⊆ Ω, S ∈ P(S)
Corollary
∀A,B such that A ⊂ B, A ⊆ B
Corollary
∀S, ∅ ⊆ S
Can we prove the latter corollary?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56
![Page 85: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/85.jpg)
Basic Definitions Two or more sets
Corollaries of subset definition
Corollary (Any set is a subset of itself)
∀S ⊆ Ω, S ⊆ S
Corollary (Any set is an element of its powerset)
∀S ⊆ Ω, S ∈ P(S)
Corollary
∀A,B such that A ⊂ B, A ⊆ B
Corollary
∀S, ∅ ⊆ S
Can we prove the latter corollary?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56
![Page 86: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/86.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ?
T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 87: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/87.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T
1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 88: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/88.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ?
F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 89: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/89.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F
1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 90: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/90.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ?
T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 91: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/91.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T
1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 92: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/92.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ?
F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 93: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/93.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F
1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 94: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/94.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ?
T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 95: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/95.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T
∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 96: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/96.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ?
T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 97: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/97.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T
∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 98: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/98.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ?
T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 99: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/99.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T
∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 100: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/100.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ?
F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 101: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/101.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F
∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 102: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/102.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ?
T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 103: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/103.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T
∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 104: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/104.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ?
F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 105: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/105.jpg)
Basic Definitions Two or more sets
Subset and membership
Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.
Are the following statements true or false?
1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F
Master those!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 29 / 56
![Page 106: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/106.jpg)
Basic Definitions Two or more sets
Union
Figure 6: Venn Diagram illustratingthe union of two sets.
Definition (Union)
Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).
Examples:
−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 30 / 56
![Page 107: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/107.jpg)
Basic Definitions Two or more sets
Union
Figure 6: Venn Diagram illustratingthe union of two sets.
Definition (Union)
Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).
Examples:
−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 30 / 56
![Page 108: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/108.jpg)
Basic Definitions Two or more sets
Union
Figure 6: Venn Diagram illustratingthe union of two sets.
Definition (Union)
Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).
Examples:
−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2
−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 30 / 56
![Page 109: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/109.jpg)
Basic Definitions Two or more sets
Union
Figure 6: Venn Diagram illustratingthe union of two sets.
Definition (Union)
Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).
Examples:
−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7
a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 30 / 56
![Page 110: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/110.jpg)
Basic Definitions Two or more sets
Union
Figure 6: Venn Diagram illustratingthe union of two sets.
Definition (Union)
Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).
Examples:
−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, c
R−Q ∪Q = R
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 30 / 56
![Page 111: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/111.jpg)
Basic Definitions Two or more sets
Union
Figure 6: Venn Diagram illustratingthe union of two sets.
Definition (Union)
Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).
Examples:
−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 30 / 56
![Page 112: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/112.jpg)
Basic Definitions Two or more sets
Corollaries of union definition
Corollary (Union is reflexive)
For every set A, A ∪A = A
Corollary (Union of a set and its absolute complement)
For every set A, A ∪A′ = Ω
Corollary (Empty set is the neutral element of union)
For every set A, A ∪ ∅ = A
Since A ∪A = A ∪ ∅ = A, why don’t we call A a neutral element ofunion as well?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 31 / 56
![Page 113: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/113.jpg)
Basic Definitions Two or more sets
Corollaries of union definition
Corollary (Union is reflexive)
For every set A, A ∪A = A
Corollary (Union of a set and its absolute complement)
For every set A, A ∪A′ = Ω
Corollary (Empty set is the neutral element of union)
For every set A, A ∪ ∅ = A
Since A ∪A = A ∪ ∅ = A, why don’t we call A a neutral element ofunion as well?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 31 / 56
![Page 114: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/114.jpg)
Basic Definitions Two or more sets
Corollaries of union definition
Corollary (Union is reflexive)
For every set A, A ∪A = A
Corollary (Union of a set and its absolute complement)
For every set A, A ∪A′ = Ω
Corollary (Empty set is the neutral element of union)
For every set A, A ∪ ∅ = A
Since A ∪A = A ∪ ∅ = A, why don’t we call A a neutral element ofunion as well?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 31 / 56
![Page 115: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/115.jpg)
Basic Definitions Two or more sets
Corollaries of union definition
Corollary (Union is reflexive)
For every set A, A ∪A = A
Corollary (Union of a set and its absolute complement)
For every set A, A ∪A′ = Ω
Corollary (Empty set is the neutral element of union)
For every set A, A ∪ ∅ = A
Since A ∪A = A ∪ ∅ = A, why don’t we call A a neutral element ofunion as well?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 31 / 56
![Page 116: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/116.jpg)
Basic Definitions Two or more sets
Intersection
Figure 7: Venn Diagram illustratingthe intersection of two sets.
Definition (Intersection)
Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).
Examples:
12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 32 / 56
![Page 117: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/117.jpg)
Basic Definitions Two or more sets
Intersection
Figure 7: Venn Diagram illustratingthe intersection of two sets.
Definition (Intersection)
Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).
Examples:
12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 32 / 56
![Page 118: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/118.jpg)
Basic Definitions Two or more sets
Intersection
Figure 7: Venn Diagram illustratingthe intersection of two sets.
Definition (Intersection)
Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).
Examples:
12,−8, 11 ∩ 11, 2 = 11
1, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 32 / 56
![Page 119: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/119.jpg)
Basic Definitions Two or more sets
Intersection
Figure 7: Venn Diagram illustratingthe intersection of two sets.
Definition (Intersection)
Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).
Examples:
12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅
∩ = ? ∅ (!)Z− ∩P = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 32 / 56
![Page 120: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/120.jpg)
Basic Definitions Two or more sets
Intersection
Figure 7: Venn Diagram illustratingthe intersection of two sets.
Definition (Intersection)
Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).
Examples:
12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)
Z− ∩P = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 32 / 56
![Page 121: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/121.jpg)
Basic Definitions Two or more sets
Intersection
Figure 7: Venn Diagram illustratingthe intersection of two sets.
Definition (Intersection)
Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).
Examples:
12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 32 / 56
![Page 122: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/122.jpg)
Basic Definitions Two or more sets
Corollaries of intersection definition
Corollary (Reflexivity of intersection)
For all sets A, A ∩A = A
Corollary (Intersection and subset)
For all sets A, B such that A ⊆ B, A ∩B = A.
Corollary (Identity law)
∀A ⊆ Ω, A ∩ Ω = A
Corollary (Domination law)
∀A ⊆ Ω, A ∩ ∅ = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 33 / 56
![Page 123: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/123.jpg)
Basic Definitions Two or more sets
Corollaries of intersection definition
Corollary (Reflexivity of intersection)
For all sets A, A ∩A = A
Corollary (Intersection and subset)
For all sets A, B such that A ⊆ B, A ∩B = A.
Corollary (Identity law)
∀A ⊆ Ω, A ∩ Ω = A
Corollary (Domination law)
∀A ⊆ Ω, A ∩ ∅ = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 33 / 56
![Page 124: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/124.jpg)
Basic Definitions Two or more sets
Corollaries of intersection definition
Corollary (Reflexivity of intersection)
For all sets A, A ∩A = A
Corollary (Intersection and subset)
For all sets A, B such that A ⊆ B, A ∩B = A.
Corollary (Identity law)
∀A ⊆ Ω, A ∩ Ω = A
Corollary (Domination law)
∀A ⊆ Ω, A ∩ ∅ = ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 33 / 56
![Page 125: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/125.jpg)
Basic Definitions Two or more sets
(Relative) Complement
Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.
Definition (Relative complement)
Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).
Examples:
12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 34 / 56
![Page 126: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/126.jpg)
Basic Definitions Two or more sets
(Relative) Complement
Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.
Definition (Relative complement)
Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).
Examples:
12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 34 / 56
![Page 127: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/127.jpg)
Basic Definitions Two or more sets
(Relative) Complement
Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.
Definition (Relative complement)
Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).
Examples:
12,−8, 11 − 11, 2 =12,−8
1, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 34 / 56
![Page 128: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/128.jpg)
Basic Definitions Two or more sets
(Relative) Complement
Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.
Definition (Relative complement)
Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).
Examples:
12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 13
0, 1 − −1, 0, 1 = ? ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 34 / 56
![Page 129: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/129.jpg)
Basic Definitions Two or more sets
(Relative) Complement
Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.
Definition (Relative complement)
Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).
Examples:
12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ?
∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 34 / 56
![Page 130: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/130.jpg)
Basic Definitions Two or more sets
(Relative) Complement
Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.
Definition (Relative complement)
Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).
Examples:
12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 34 / 56
![Page 131: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/131.jpg)
Basic Definitions Two or more sets
Disjoint sets
Definition (Disjoint sets)
Two sets A and B are called disjoint if and only if A ∩B = ∅
Corollary
∀A ⊆ Ω, A and ∅ are disjoint.
Corollary
∀A ⊆ Ω, A and A′ are disjoint.
Corollary
For all sets A, B, A−B and B are disjoint.
Corollary
∅ is the only set disjoint from Ω.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 35 / 56
![Page 132: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/132.jpg)
Basic Definitions Two or more sets
Disjoint sets
Definition (Disjoint sets)
Two sets A and B are called disjoint if and only if A ∩B = ∅
Corollary
∀A ⊆ Ω, A and ∅ are disjoint.
Corollary
∀A ⊆ Ω, A and A′ are disjoint.
Corollary
For all sets A, B, A−B and B are disjoint.
Corollary
∅ is the only set disjoint from Ω.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 35 / 56
![Page 133: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/133.jpg)
Basic Definitions Two or more sets
Disjoint sets
Definition (Disjoint sets)
Two sets A and B are called disjoint if and only if A ∩B = ∅
Corollary
∀A ⊆ Ω, A and ∅ are disjoint.
Corollary
∀A ⊆ Ω, A and A′ are disjoint.
Corollary
For all sets A, B, A−B and B are disjoint.
Corollary
∅ is the only set disjoint from Ω.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 35 / 56
![Page 134: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/134.jpg)
Basic Definitions Two or more sets
Disjoint sets
Definition (Disjoint sets)
Two sets A and B are called disjoint if and only if A ∩B = ∅
Corollary
∀A ⊆ Ω, A and ∅ are disjoint.
Corollary
∀A ⊆ Ω, A and A′ are disjoint.
Corollary
For all sets A, B, A−B and B are disjoint.
Corollary
∅ is the only set disjoint from Ω.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 35 / 56
![Page 135: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/135.jpg)
Basic Definitions Two or more sets
Disjoint sets
Definition (Disjoint sets)
Two sets A and B are called disjoint if and only if A ∩B = ∅
Corollary
∀A ⊆ Ω, A and ∅ are disjoint.
Corollary
∀A ⊆ Ω, A and A′ are disjoint.
Corollary
For all sets A, B, A−B and B are disjoint.
Corollary
∅ is the only set disjoint from Ω.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 35 / 56
![Page 136: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/136.jpg)
Basic Definitions Two or more sets
Partition
Figure 9: Venn Diagram illustrating apartition of a set.
Definition (Partition of a set)
Let A be a set. The partition of Ais a set of sets A1, A2, . . . , An whichhave the following properties:
1 Ai and Aj are disjoint forevery i 6= j, 1 ≤ i, j ≤ n
2⋃nk=1Ak = A.
Examples:
R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 36 / 56
![Page 137: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/137.jpg)
Basic Definitions Two or more sets
Partition
Figure 9: Venn Diagram illustrating apartition of a set.
Definition (Partition of a set)
Let A be a set. The partition of Ais a set of sets A1, A2, . . . , An whichhave the following properties:
1 Ai and Aj are disjoint forevery i 6= j, 1 ≤ i, j ≤ n
2⋃nk=1Ak = A.
Examples:
R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 36 / 56
![Page 138: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/138.jpg)
Basic Definitions Two or more sets
Partition
Figure 9: Venn Diagram illustrating apartition of a set.
Definition (Partition of a set)
Let A be a set. The partition of Ais a set of sets A1, A2, . . . , An whichhave the following properties:
1 Ai and Aj are disjoint forevery i 6= j, 1 ≤ i, j ≤ n
2⋃nk=1Ak = A.
Examples:
R and Z−, 0,N+
−10,−5, 6 and−10,−5, 6 .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 36 / 56
![Page 139: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/139.jpg)
Basic Definitions Two or more sets
Partition
Figure 9: Venn Diagram illustrating apartition of a set.
Definition (Partition of a set)
Let A be a set. The partition of Ais a set of sets A1, A2, . . . , An whichhave the following properties:
1 Ai and Aj are disjoint forevery i 6= j, 1 ≤ i, j ≤ n
2⋃nk=1Ak = A.
Examples:
R and Z−, 0,N+−10,−5, 6 and−10,−5, 6
.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 36 / 56
![Page 140: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/140.jpg)
Basic Definitions Two or more sets
Partition
Figure 9: Venn Diagram illustrating apartition of a set.
Definition (Partition of a set)
Let A be a set. The partition of Ais a set of sets A1, A2, . . . , An whichhave the following properties:
1 Ai and Aj are disjoint forevery i 6= j, 1 ≤ i, j ≤ n
2⋃nk=1Ak = A.
Examples:
R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 36 / 56
![Page 141: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/141.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 142: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/142.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 143: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/143.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ?
YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 144: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/144.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES
2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 145: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/145.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ?
YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 146: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/146.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES
3 (−8√
2,−16,−7, 0, 1) ? NO4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 147: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/147.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ?
NO4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 148: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/148.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 149: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/149.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ?
YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 150: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/150.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES
5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 151: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/151.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ?
YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 152: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/152.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)
6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 153: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/153.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ?
YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 154: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/154.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES
7 (0, 0.5,√
2, 3, 4,√
9) NO8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 155: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/155.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9)
NO8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 156: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/156.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 157: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/157.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . )
NO
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 158: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/158.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NOJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 159: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/159.jpg)
Basic Definitions Two or more sets
Ordered n-tuples
Definition (Ordered n-tuple)
Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.
aE.g a total order.
Definition (n-tuple equality)
Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:
1 n = r.
2 ∀i = 1, 2, . . . , n, xi = yi.
Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8
√2,−16,−7, 0, 1) ? NO
4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,
√2, 3, 4,
√9) NO
8 (0, 1, 2, . . . ) NOJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56
![Page 160: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/160.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 161: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/161.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 162: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/162.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)
Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 163: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/163.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)
N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 164: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/164.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 165: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/165.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 166: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/166.jpg)
Basic Definitions Two or more sets
Cartesian Product
Definition (Cartesian Product)
Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B
Examples:
0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .
Order matters! (A×B) 6= (B ×A)
The elements of the Cartesian Product are ordered n-tuples!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 38 / 56
![Page 167: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/167.jpg)
Proofs with sets
Proofs with sets
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 39 / 56
![Page 168: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/168.jpg)
Proofs with sets
Proving subset relationships
One needs to prove that whenever an element belongs to a set x, itmust belong to the other.
Examples:
N ⊆ Qs | s is a student registered in 250 ⊆ UMD StudentsKansas Counties ⊆ USA Counties
What kinds of proofs are required here?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 40 / 56
![Page 169: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/169.jpg)
Proofs with sets
Proving subset relationships
One needs to prove that whenever an element belongs to a set x, itmust belong to the other.
Examples:
N ⊆ Qs | s is a student registered in 250 ⊆ UMD StudentsKansas Counties ⊆ USA Counties
What kinds of proofs are required here?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 40 / 56
![Page 170: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/170.jpg)
Proofs with sets
Proving subset relationships
One needs to prove that whenever an element belongs to a set x, itmust belong to the other.
Examples:
N ⊆ Qs | s is a student registered in 250 ⊆ UMD StudentsKansas Counties ⊆ USA Counties
What kinds of proofs are required here?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 40 / 56
![Page 171: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/171.jpg)
Proofs with sets
An example proof
Theorem
For any sets A and B, A ∩B ⊆ A
Proof.
Let M , N be generic particular sets. Then, by the definition ofintersection, M ∩N = x|x ∈M ∧ x ∈ N . So all those elements xbelong in M as well, which means that M ∩N ⊆M by the definitionof subset. Since M and N were chosen arbitrarily, the result holds forevery pair of sets A,B.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 41 / 56
![Page 172: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/172.jpg)
Proofs with sets
An example proof
Theorem
For any sets A and B, A ∩B ⊆ A
Proof.
Let M , N be generic particular sets. Then, by the definition ofintersection, M ∩N = x|x ∈M ∧ x ∈ N . So all those elements xbelong in M as well, which means that M ∩N ⊆M by the definitionof subset. Since M and N were chosen arbitrarily, the result holds forevery pair of sets A,B.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 41 / 56
![Page 173: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/173.jpg)
Proofs with sets
Proving equality relationships
Definition (Set equality)
Let A and B be sets. Then, A and B are equal, denoted A = B, if andonly if A ⊆ B and B ⊆ A.
So we need to prove the subset relationship both ways.
Examples:
Let A = Neven and B = (Z− Zodd) ∩ R+ Prove that A = B.Let A = n2 |n is odd and Zodd. Is A = B?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 42 / 56
![Page 174: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/174.jpg)
Proofs with sets
Proving equality relationships
Definition (Set equality)
Let A and B be sets. Then, A and B are equal, denoted A = B, if andonly if A ⊆ B and B ⊆ A.
So we need to prove the subset relationship both ways.
Examples:
Let A = Neven and B = (Z− Zodd) ∩ R+ Prove that A = B.Let A = n2 |n is odd and Zodd. Is A = B?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 42 / 56
![Page 175: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/175.jpg)
Proofs with sets
Proving equality relationships
Definition (Set equality)
Let A and B be sets. Then, A and B are equal, denoted A = B, if andonly if A ⊆ B and B ⊆ A.
So we need to prove the subset relationship both ways.
Examples:
Let A = Neven and B = (Z− Zodd) ∩ R+ Prove that A = B.Let A = n2 |n is odd and Zodd. Is A = B?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 42 / 56
![Page 176: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/176.jpg)
Proofs with sets
Proving equality relationships
Definition (Set equality)
Let A and B be sets. Then, A and B are equal, denoted A = B, if andonly if A ⊆ B and B ⊆ A.
So we need to prove the subset relationship both ways.
Examples:
Let A = Neven and B = (Z− Zodd) ∩ R+ Prove that A = B.Let A = n2 |n is odd and Zodd. Is A = B?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 42 / 56
![Page 177: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/177.jpg)
Proofs with sets
Take 5
Let’s split into teams and try to prove the following:1 A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)2 (A ∪B)′ = A′ ∩B′3 A ∪ (A ∩B) = A
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 43 / 56
![Page 178: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/178.jpg)
Proofs with sets
Proving that a set is empty
Usually done via contradiction.
Assume it is non-empty, so it must contain some element x, reacha contradiction.
Prove that Zeven and Zodd are disjoint for practice.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 44 / 56
![Page 179: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/179.jpg)
Proofs with sets
Axioms of Classical Set Theory
For sets A, B and universal domain Ω:Commutativity A ∪B = B ∪A A ∩B = B ∩AAssociativity of union& intersection
(A ∩B) ∩ C = A ∩ (B ∩ C) (A ∪B) ∪ C = A ∪ (B ∪ C)
Distributivity ofunion & intersection
A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)
Identity laws A ∩ Ω = A A ∪ ∅ = AInverse laws A ∪A′ = Ω A ∩A′ = ∅Double complemen-tation
(A′)′ = A
Idempotence A ∩A = A A ∪A = ADe Morgan’s axioms (A ∩B)′ = A′ ∪B′ (A ∪B)′ = A′ ∩B′
Universal bound(Domination) laws
A ∪ Ω = Ω A ∩ ∅ = ∅
Absorption laws A ∪ (A ∩B) = A A ∩ (A ∪B) = AAbsolute Comple-ments of empty set /domain
∅′ = Ω Ω′ = ∅
Unnamed #1 A ⊆ B ⇒ A ∩B = A A ⊆ A ∪B = BUnnamed #2 A ⊆ B ⇒ B′
Unnamed #3 A⊕B = A ∪B −A ∩B
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 45 / 56
![Page 180: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/180.jpg)
Proofs with sets
Applying the axioms
Now that we have the axioms of set theory set up, we can usethem to derive new relationships!
Let’s use the axioms to prove the following:
((A1 ∪A2) ∪A3) ∪A4 = A1 ∪ ((A2 ∪A3) ∪A4)(A ∪B)− C = (A− C) ∪ (B − C)
As with the propositional logic exercises, be meticulous!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 46 / 56
![Page 181: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/181.jpg)
An application: Formal languages
An application: Formal languages
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 47 / 56
![Page 182: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/182.jpg)
An application: Formal languages
Alphabets
Any finite set can be considered an alphabet (sometimes called avocabulary).
Examples:
English alphabet: a, b, c, . . . , zGreek alphabet: α, β, . . . , ωBinary alphabet: 0, 1
Denoted Σ.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 48 / 56
![Page 183: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/183.jpg)
An application: Formal languages
Alphabets
Any finite set can be considered an alphabet (sometimes called avocabulary).
Examples:
English alphabet: a, b, c, . . . , zGreek alphabet: α, β, . . . , ωBinary alphabet: 0, 1
Denoted Σ.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 48 / 56
![Page 184: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/184.jpg)
An application: Formal languages
Alphabets
Any finite set can be considered an alphabet (sometimes called avocabulary).
Examples:
English alphabet: a, b, c, . . . , zGreek alphabet: α, β, . . . , ωBinary alphabet: 0, 1
Denoted Σ.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 48 / 56
![Page 185: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/185.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 186: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/186.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):
acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 187: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/187.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):a
caaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 188: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/188.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):ac
aaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 189: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/189.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaa
bababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 190: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/190.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababaz
jasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 191: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/191.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjason
madagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 192: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/192.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascar
charliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 193: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/193.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharlie
thissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 194: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/194.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishletters
ε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 195: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/195.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 196: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/196.jpg)
An application: Formal languages
Strings
Any ordered n-tuple of symbols over an alphabet Σ.
Usually denoted σ.
Length of a string: `(σ) = the number of “characters” in σ.
Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string
`(ε) = 0.
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 49 / 56
![Page 197: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/197.jpg)
An application: Formal languages
Σn, Σ∗ and Languages
Are we straying away from sets or something?
Nope.
Some additional definitions:
Σn, for n ∈ N: The set of strings made up of elements of Σ withlength exactly n.Σ∗: The set of all strings made up of elements of Σ with finitelength.Language L over Σ: any subset of Σ∗.
Examples (whiteboard)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 50 / 56
![Page 198: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/198.jpg)
An application: Formal languages
Σn, Σ∗ and Languages
Are we straying away from sets or something?
Nope.
Some additional definitions:
Σn, for n ∈ N: The set of strings made up of elements of Σ withlength exactly n.Σ∗: The set of all strings made up of elements of Σ with finitelength.Language L over Σ: any subset of Σ∗.
Examples (whiteboard)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 50 / 56
![Page 199: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/199.jpg)
An application: Formal languages
Σn, Σ∗ and Languages
Are we straying away from sets or something?
Nope.
Some additional definitions:
Σn, for n ∈ N: The set of strings made up of elements of Σ withlength exactly n.Σ∗: The set of all strings made up of elements of Σ with finitelength.Language L over Σ: any subset of Σ∗.
Examples (whiteboard)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 50 / 56
![Page 200: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/200.jpg)
An application: Formal languages
Σn, Σ∗ and Languages
Are we straying away from sets or something?
Nope.
Some additional definitions:
Σn, for n ∈ N: The set of strings made up of elements of Σ withlength exactly n.Σ∗: The set of all strings made up of elements of Σ with finitelength.Language L over Σ: any subset of Σ∗.
Examples (whiteboard)
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 50 / 56
![Page 201: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/201.jpg)
Paradoxes in Set Theory
Paradoxes in Set Theory
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 51 / 56
![Page 202: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/202.jpg)
Paradoxes in Set Theory
The barber paradox
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 52 / 56
![Page 203: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/203.jpg)
Paradoxes in Set Theory
The barber paradox
In rural Cleanshaveville, it is illegal for guys to have facial hair.
To accomplish this, the town employs a single barber.
The barber is tasked with shaving those, and only those menwho do not shave themselves (say, at home).
Question: Who shaves the barber?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 53 / 56
![Page 204: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/204.jpg)
Paradoxes in Set Theory
The barber paradox
In rural Cleanshaveville, it is illegal for guys to have facial hair.
To accomplish this, the town employs a single barber.
The barber is tasked with shaving those, and only those menwho do not shave themselves (say, at home).
Question: Who shaves the barber?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 53 / 56
![Page 205: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/205.jpg)
Paradoxes in Set Theory
The barber paradox
In rural Cleanshaveville, it is illegal for guys to have facial hair.
To accomplish this, the town employs a single barber.
The barber is tasked with shaving those, and only those menwho do not shave themselves (say, at home).
Question: Who shaves the barber?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 53 / 56
![Page 206: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/206.jpg)
Paradoxes in Set Theory
Russel’s paradox
We already know that a set can have sets as its elements.
Let’s define a set S as the set of sets that are not membersof themselves.
Symbolically: S = x | x is a set such that x /∈ xQuestion: Is S ∈ S?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 54 / 56
![Page 207: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/207.jpg)
Paradoxes in Set Theory
Russel’s paradox
We already know that a set can have sets as its elements.
Let’s define a set S as the set of sets that are not membersof themselves.
Symbolically: S = x | x is a set such that x /∈ xQuestion: Is S ∈ S?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 54 / 56
![Page 208: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/208.jpg)
Paradoxes in Set Theory
Russel’s paradox
We already know that a set can have sets as its elements.
Let’s define a set S as the set of sets that are not membersof themselves.
Symbolically: S = x | x is a set such that x /∈ x
Question: Is S ∈ S?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 54 / 56
![Page 209: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/209.jpg)
Paradoxes in Set Theory
Russel’s paradox
We already know that a set can have sets as its elements.
Let’s define a set S as the set of sets that are not membersof themselves.
Symbolically: S = x | x is a set such that x /∈ xQuestion: Is S ∈ S?
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 54 / 56
![Page 210: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/210.jpg)
Paradoxes in Set Theory
The Halting Problem
Perhaps our most important result.
Demonstrated by Alan Turing.
Question: Can I write a computer program that takesanother program as input and tells me if it terminates (infinite time) or not given some input?
Symbolically, our program would look like this:
PROGRAM HALTS(P , x)INPUTS: P , a program, x, a string fed as input to P .OUTPUTS: YES, if P terminates, NO, otherwise.BEGIN
Run P on xIf P terminates, output YESelse, output NO
END
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 55 / 56
![Page 211: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/211.jpg)
Paradoxes in Set Theory
The Halting Problem
Perhaps our most important result.
Demonstrated by Alan Turing.
Question: Can I write a computer program that takesanother program as input and tells me if it terminates (infinite time) or not given some input?
Symbolically, our program would look like this:
PROGRAM HALTS(P , x)INPUTS: P , a program, x, a string fed as input to P .OUTPUTS: YES, if P terminates, NO, otherwise.BEGIN
Run P on xIf P terminates, output YESelse, output NO
END
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 55 / 56
![Page 212: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/212.jpg)
Paradoxes in Set Theory
The Halting Problem
Perhaps our most important result.
Demonstrated by Alan Turing.
Question: Can I write a computer program that takesanother program as input and tells me if it terminates (infinite time) or not given some input?
Symbolically, our program would look like this:
PROGRAM HALTS(P , x)INPUTS: P , a program, x, a string fed as input to P .OUTPUTS: YES, if P terminates, NO, otherwise.BEGIN
Run P on xIf P terminates, output YESelse, output NO
ENDJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 55 / 56
![Page 213: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/213.jpg)
Paradoxes in Set Theory
The Halting Problem
Suppose that I can write HALTS.
Then, it is also possible to write a program calledHALTS TESTER, as follows:
PROGRAM HALTS TESTER(P )INPUTS: A program POUTPUTS: YES or nothing (see below)BEGIN:
If HALTS(P, P) outputs YES, then loop foreverelse, output YES
END
Paradox.
Upshot: The program HALTS does not exist!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 56 / 56
![Page 214: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/214.jpg)
Paradoxes in Set Theory
The Halting Problem
Suppose that I can write HALTS.
Then, it is also possible to write a program calledHALTS TESTER, as follows:
PROGRAM HALTS TESTER(P )INPUTS: A program POUTPUTS: YES or nothing (see below)BEGIN:
If HALTS(P, P) outputs YES, then loop foreverelse, output YES
END
Paradox.
Upshot: The program HALTS does not exist!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 56 / 56
![Page 215: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/215.jpg)
Paradoxes in Set Theory
The Halting Problem
Suppose that I can write HALTS.
Then, it is also possible to write a program calledHALTS TESTER, as follows:
PROGRAM HALTS TESTER(P )INPUTS: A program POUTPUTS: YES or nothing (see below)BEGIN:
If HALTS(P, P) outputs YES, then loop foreverelse, output YES
END
Paradox.
Upshot: The program HALTS does not exist!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 56 / 56
![Page 216: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/216.jpg)
Paradoxes in Set Theory
The Halting Problem
Suppose that I can write HALTS.
Then, it is also possible to write a program calledHALTS TESTER, as follows:
PROGRAM HALTS TESTER(P )INPUTS: A program POUTPUTS: YES or nothing (see below)BEGIN:
If HALTS(P, P) outputs YES, then loop foreverelse, output YES
END
Paradox.
Upshot: The program HALTS does not exist!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 56 / 56
![Page 217: Set Theory - UMD Department of Computer Science · 2016-07-13 · Outline 1 Branches of Set Theory 2 Basic De nitions Single sets Two or more sets 3 Proofs with sets 4 An application:](https://reader034.vdocument.in/reader034/viewer/2022042913/5f4c6c8a99ad303c8c38d60b/html5/thumbnails/217.jpg)
Paradoxes in Set Theory
The Halting Problem
Suppose that I can write HALTS.
Then, it is also possible to write a program calledHALTS TESTER, as follows:
PROGRAM HALTS TESTER(P )INPUTS: A program POUTPUTS: YES or nothing (see below)BEGIN:
If HALTS(P, P) outputs YES, then loop foreverelse, output YES
END
Paradox.
Upshot: The program HALTS does not exist!
Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 56 / 56