set theory - umd department of computer science · 2016-07-13 · outline 1 branches of set theory...

Set Theory

Jason Filippou

CMSC250 @ UMCP

06-20-2016

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 1 / 56

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

Branches of Set Theory

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!

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!

Based entirely on Venn Diagrams.

Α ΒC

Figure 1: An example Venn Diagram.

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

Famous Result

1874 Cantor paper: “On a Property of the Collection of All RealAlgebraic Numbers”

The set of real numbers is uncountable.

Power set operation.Cantor’s paradise.

Famous Result

1874 Cantor paper: “On a Property of the Collection of All RealAlgebraic Numbers”

The set of real numbers is uncountable.

Power set operation.Cantor’s paradise.

Russel’s Paradox

Consider the following set:

S = x|x /∈ x

Then, does S ∈ S ?

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.

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

Kripke-Platek

Morse-Kelley

Tarski-Grothendieck

Kripke-Platek

Morse-Kelley

Tarski-Grothendieck

Kripke-Platek

Morse-Kelley

Tarski-Grothendieck

Kripke-Platek

Morse-Kelley

Tarski-Grothendieck

Kripke-Platek

Morse-Kelley

Tarski-Grothendieck

Basic Definitions

Basic Definitions Single sets

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.

Definitions

Definition (Set)

Definitions

Definition (Set)

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?

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?

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)

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, . . .

Defining a set

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)

Defining a set

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)

Defining a set

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 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ?

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ?

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ?

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ?

ℵ0 (???)

Cardinality

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)

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....

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

The empty set

Corollary

|∅| = 0

Corollary

∀A ⊆ Ω, ∅ ⊆ A

The empty set

Corollary

|∅| = 0

Corollary

∀A ⊆ Ω, ∅ ⊆ A

Some practice

How many elements do the following sets contain?

1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅

2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅2 ∅

3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅2 ∅3

4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . .

5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . .

6 . . . 250 . . . 7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . .

7 ∅, ∅

Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅

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)

The powerset

Corollary

∀A ⊆ Ω, ∅ ∈ P(A)

Corollary

∀A ⊆ Ω, A ∈ P(A)

The powerset

Corollary

∀A ⊆ Ω, ∅ ∈ P(A)

Corollary

∀A ⊆ Ω, A ∈ P(A)

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

(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′ = ∅

(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′ = ∅

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.

Caution: Sets and intervals

This: 0, 1, 2, . . . , 10Is not the same as this: [0, . . . , 10].

Intervals are sets, but sets are not intervals.

Exercises

Provide the following sets in interval or set notation. Assume thatΩ = R

(Z∗−)′ = ?(−1, 1] ∪ −1 = ?[0, 10]′ = ?

Basic Definitions Two or more sets

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.

Subset

Definition (Subset)

Examples:

N ⊆ Z

P ⊆ Qa ∈ Z

∣∣4|a ⊆ Zeven

Subset

Definition (Subset)

Examples:

N ⊆ ZP ⊆ Q

a ∈ Z∣∣4|a ⊆ Zeven

Subset

Definition (Subset)

Examples:

∣∣4|a ⊆ Zeven

Subset

Definition (Subset)

Examples:

∣∣4|a ⊆ Zeven

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.

Proper subset

Examples:

Q ⊂ R

1 ⊂ 0, 1 ⊂ ∅, ∅

Proper subset

Examples:

Q ⊂ R1 ⊂ 0, 1

⊂ ∅, ∅

Proper subset

Examples:

Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅

Proper subset

Examples:

Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅

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?

∀S ⊆ Ω, S ⊆ S

∀S ⊆ Ω, S ∈ P(S)

Corollary

∀S, ∅ ⊆ S

∀S ⊆ Ω, S ⊆ S

∀S ⊆ Ω, S ∈ P(S)

Corollary

∀S, ∅ ⊆ S

∀S ⊆ Ω, S ⊆ S

∀S ⊆ Ω, S ∈ P(S)

Corollary

∀S, ∅ ⊆ S

∀S ⊆ Ω, S ⊆ S

∀S ⊆ Ω, S ∈ P(S)

Corollary

∀S, ∅ ⊆ S

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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!

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∅ ∈ ∅ ?

Master those!

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!

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

Definition (Union)

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

Definition (Union)

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

Definition (Union)

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

Definition (Union)

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

Definition (Union)

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

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?

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 = ∅

Intersection

Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅

Intersection

Examples:

12,−8, 11 ∩ 11, 2 = 11

1, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅

Intersection

Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅

∩ = ? ∅ (!)Z− ∩P = ∅

Intersection

Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)

Z− ∩P = ∅

Intersection

Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅

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 ∩ ∅ = ∅

∀A ⊆ Ω, A ∩ Ω = A

∀A ⊆ Ω, A ∩ ∅ = ∅

∀A ⊆ Ω, A ∩ Ω = A

∀A ⊆ Ω, A ∩ ∅ = ∅

(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 = ? ∅

Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅

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 = ? ∅

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 = ? ∅

Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ?

Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅

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.

Disjoint sets

Corollary

Disjoint sets

Corollary

Disjoint sets

Corollary

Disjoint sets

Corollary

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 .

Partition

2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .

Partition

2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+

−10,−5, 6 and−10,−5, 6 .

Partition

2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+−10,−5, 6 and−10,−5, 6

Partition

2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .

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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

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

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

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

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

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

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√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

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

NO8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . )

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NOJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56

Ordered n-tuples

aE.g a total order.

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

√2,−16,−7, 0, 1) ? NO

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NOJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56

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!

Cartesian Product

Examples:

Cartesian Product

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), . . .

Cartesian Product

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), . . .

Cartesian Product

Examples:

Cartesian Product

Examples:

Cartesian Product

Examples:

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?

Proofs with sets

Examples:

Proofs with sets

Examples:

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.

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.

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?

Proofs with sets

Examples:

Proofs with sets

Examples:

Proofs with sets

Examples:

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

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.

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

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!

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.

Alphabets

Examples:

Denoted Σ.

Alphabets

Examples:

Denoted Σ.

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.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):

acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):a

caaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):ac

aaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaa

bababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababaz

jasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjason

madagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascar

charliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharlie

thissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishletters

ε: The empty string

`(ε) = 0.

Strings

Usually denoted σ.

`(ε) = 0.

Strings

Usually denoted σ.

`(ε) = 0.

Σn, Σ∗ and Languages

Are we straying away from sets or something?

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)

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?

The barber paradox

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?

Russel’s paradox

Symbolically: S = x | x is a set such that x /∈ x

Question: Is S ∈ S?

Russel’s paradox

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

The Halting Problem

ENDJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 55 / 56

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

Paradox.

Upshot: The program HALTS does not exist!

The Halting Problem

Paradox.

The Halting Problem

Paradox.

The Halting Problem

Paradox.

The Halting Problem

Paradox.

set theory - umd department of computer science · 2016-07-13 · outline 1 branches of set theory...

Documents

introduction to set theory -...

set theory (part ii) counting principles for the union and...

set theory - half hollow hills central school district...set...

set theory notes · 2013. 11. 18. · 1 set theory notes on...

fa.its.tudelft.nl · 25. descriptive set theory descriptive...

set theory - log...

set theory topic 2: sets and set notation. i can provide...

math1081 revision - set theory, number theory and graph...

931102fuzzy set theory chap07.ppt1 fuzzy relations fuzzy...

theory of set · this unit aims at explaining the set...

introduction to set theory. ways of describing sets

set theory ab c. this lecture we will first introduce set...

properties and relationships of set theory. properties and...

chapter 1 set theory. chapter 1 1.1 types of sets and set...

mat1202 set theory: week 1 · introduction history of set...

basic set theory - american mathematical societybasic set...

basic set theory - alexandru ioan cuza...

chapter 3: set theory and logic. 3.1 types of sets and set...

logic & set theory - philadelphia university · logic & set...

james tam basic set theory you will learn basic properties...