discrete mathematics - predicates and sets
DESCRIPTION
Predicates, quantifiers. Sets, subsets, power sets, set operations, laws of set theory, principle of inclusion-exclusion.TRANSCRIPT
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Discrete MathematicsPredicates and Sets
H. Turgut Uyar Aysegul Gencata Yayımlı Emre Harmancı
2001-2013
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License
c©2001-2013 T. Uyar, A. Yayımlı, E. Harmancı
You are free:
to Share – to copy, distribute and transmit the work
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Under the following conditions:
Attribution – You must attribute the work in the manner specified by the author or licensor (but not in anyway that suggests that they endorse you or your use of the work).
Noncommercial – You may not use this work for commercial purposes.
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Legal code (the full license):http://creativecommons.org/licenses/by-nc-sa/3.0/
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Predicate
Definition
predicate (or open statement): a declarative sentence which
contains one or more variables, and
is not a proposition, but
becomes a proposition when the variables in itare replaced by certain allowable choices
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Universe of Discourse
Definition
universe of discourse: Uset of allowable choices
examples:
Z: integersN: natural numbersZ+: positive integersQ: rational numbersR: real numbersC: complex numbers
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Universe of Discourse
Definition
universe of discourse: Uset of allowable choices
examples:
Z: integersN: natural numbersZ+: positive integersQ: rational numbersR: real numbersC: complex numbers
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Predicate Examples
Example
U = Np(x): x + 2 is an even integer
p(5): Fp(8): T
¬p(x): x + 2 is not an even integer
Example
U = Nq(x , y): x + y and x − 2y are even integers
q(11, 3): F , q(14, 4): T
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Predicate Examples
Example
U = Np(x): x + 2 is an even integer
p(5): Fp(8): T
¬p(x): x + 2 is not an even integer
Example
U = Nq(x , y): x + y and x − 2y are even integers
q(11, 3): F , q(14, 4): T
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Predicate Examples
Example
U = Np(x): x + 2 is an even integer
p(5): Fp(8): T
¬p(x): x + 2 is not an even integer
Example
U = Nq(x , y): x + y and x − 2y are even integers
q(11, 3): F , q(14, 4): T
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Quantifiers
Definition
existential quantifier:predicate is true for some values
symbol: ∃read: there exists
symbol: ∃!read: there exists only one
Definition
universal quantifier:predicate is true for all values
symbol: ∀read: for all
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Quantifiers
Definition
existential quantifier:predicate is true for some values
symbol: ∃read: there exists
symbol: ∃!read: there exists only one
Definition
universal quantifier:predicate is true for all values
symbol: ∀read: for all
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Quantifiers
Definition
existential quantifier:predicate is true for some values
symbol: ∃read: there exists
symbol: ∃!read: there exists only one
Definition
universal quantifier:predicate is true for all values
symbol: ∀read: for all
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Quantifiers
existential quantifier
U = {x1, x2, . . . , xn}∃x p(x) ≡ p(x1) ∨ p(x2) ∨ · · · ∨ p(xn)
p(x) is true for some x
universal quantifier
U = {x1, x2, . . . , xn}∀x p(x) ≡ p(x1) ∧ p(x2) ∧ · · · ∧ p(xn)
p(x) is true for all x
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Quantifiers
existential quantifier
U = {x1, x2, . . . , xn}∃x p(x) ≡ p(x1) ∨ p(x2) ∨ · · · ∨ p(xn)
p(x) is true for some x
universal quantifier
U = {x1, x2, . . . , xn}∀x p(x) ≡ p(x1) ∧ p(x2) ∧ · · · ∧ p(xn)
p(x) is true for all x
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Quantifier Examples
Example
U = R
p(x) : x ≥ 0
q(x) : x2 ≥ 0
r(x) : (x − 4)(x + 1) = 0
s(x) : x2 − 3 > 0
are the following expressions true?
∃x [p(x) ∧ r(x)]
∀x [p(x) → q(x)]
∀x [q(x) → s(x)]
∀x [r(x) ∨ s(x)]
∀x [r(x) → p(x)]
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Quantifier Examples
Example
U = R
p(x) : x ≥ 0
q(x) : x2 ≥ 0
r(x) : (x − 4)(x + 1) = 0
s(x) : x2 − 3 > 0
are the following expressions true?
∃x [p(x) ∧ r(x)]
∀x [p(x) → q(x)]
∀x [q(x) → s(x)]
∀x [r(x) ∨ s(x)]
∀x [r(x) → p(x)]
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Quantifier Examples
Example
U = R
p(x) : x ≥ 0
q(x) : x2 ≥ 0
r(x) : (x − 4)(x + 1) = 0
s(x) : x2 − 3 > 0
are the following expressions true?
∃x [p(x) ∧ r(x)]
∀x [p(x) → q(x)]
∀x [q(x) → s(x)]
∀x [r(x) ∨ s(x)]
∀x [r(x) → p(x)]
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Quantifier Examples
Example
U = R
p(x) : x ≥ 0
q(x) : x2 ≥ 0
r(x) : (x − 4)(x + 1) = 0
s(x) : x2 − 3 > 0
are the following expressions true?
∃x [p(x) ∧ r(x)]
∀x [p(x) → q(x)]
∀x [q(x) → s(x)]
∀x [r(x) ∨ s(x)]
∀x [r(x) → p(x)]
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Quantifier Examples
Example
U = R
p(x) : x ≥ 0
q(x) : x2 ≥ 0
r(x) : (x − 4)(x + 1) = 0
s(x) : x2 − 3 > 0
are the following expressions true?
∃x [p(x) ∧ r(x)]
∀x [p(x) → q(x)]
∀x [q(x) → s(x)]
∀x [r(x) ∨ s(x)]
∀x [r(x) → p(x)]
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Quantifier Examples
Example
U = R
p(x) : x ≥ 0
q(x) : x2 ≥ 0
r(x) : (x − 4)(x + 1) = 0
s(x) : x2 − 3 > 0
are the following expressions true?
∃x [p(x) ∧ r(x)]
∀x [p(x) → q(x)]
∀x [q(x) → s(x)]
∀x [r(x) ∨ s(x)]
∀x [r(x) → p(x)]
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Negating Quantifiers
replace ∀ with ∃, and ∃ with ∀negate the predicate
¬∃x p(x) ⇔ ∀x ¬p(x)
¬∃x ¬p(x) ⇔ ∀x p(x)
¬∀x p(x) ⇔ ∃x ¬p(x)
¬∀x ¬p(x) ⇔ ∃x p(x)
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Negating Quantifiers
replace ∀ with ∃, and ∃ with ∀negate the predicate
¬∃x p(x) ⇔ ∀x ¬p(x)
¬∃x ¬p(x) ⇔ ∀x p(x)
¬∀x p(x) ⇔ ∃x ¬p(x)
¬∀x ¬p(x) ⇔ ∃x p(x)
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Negating Quantifiers
Theorem
¬∃x p(x) ⇔ ∀x ¬p(x)
Proof.
¬∃x p(x) ≡ ¬[p(x1) ∨ p(x2) ∨ · · · ∨ p(xn)]
⇔ ¬p(x1) ∧ ¬p(x2) ∧ · · · ∧ ¬p(xn)
≡ ∀x ¬p(x)
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Negating Quantifiers
Theorem
¬∃x p(x) ⇔ ∀x ¬p(x)
Proof.
¬∃x p(x) ≡ ¬[p(x1) ∨ p(x2) ∨ · · · ∨ p(xn)]
⇔ ¬p(x1) ∧ ¬p(x2) ∧ · · · ∧ ¬p(xn)
≡ ∀x ¬p(x)
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Negating Quantifiers
Theorem
¬∃x p(x) ⇔ ∀x ¬p(x)
Proof.
¬∃x p(x) ≡ ¬[p(x1) ∨ p(x2) ∨ · · · ∨ p(xn)]
⇔ ¬p(x1) ∧ ¬p(x2) ∧ · · · ∧ ¬p(xn)
≡ ∀x ¬p(x)
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Negating Quantifiers
Theorem
¬∃x p(x) ⇔ ∀x ¬p(x)
Proof.
¬∃x p(x) ≡ ¬[p(x1) ∨ p(x2) ∨ · · · ∨ p(xn)]
⇔ ¬p(x1) ∧ ¬p(x2) ∧ · · · ∧ ¬p(xn)
≡ ∀x ¬p(x)
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Predicate Equivalences
Theorem
∃x [p(x) ∨ q(x)] ⇔ ∃x p(x) ∨ ∃x q(x)
Theorem
∀x [p(x) ∧ q(x)] ⇔ ∀x p(x) ∧ ∀x q(x)
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Predicate Equivalences
Theorem
∃x [p(x) ∨ q(x)] ⇔ ∃x p(x) ∨ ∃x q(x)
Theorem
∀x [p(x) ∧ q(x)] ⇔ ∀x p(x) ∧ ∀x q(x)
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Predicate Implications
Theorem
∀x p(x) ⇒ ∃x p(x)
Theorem
∃x [p(x) ∧ q(x)] ⇒ ∃x p(x) ∧ ∃x q(x)
Theorem
∀x p(x) ∨ ∀x q(x) ⇒ ∀x [p(x) ∨ q(x)]
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Predicate Implications
Theorem
∀x p(x) ⇒ ∃x p(x)
Theorem
∃x [p(x) ∧ q(x)] ⇒ ∃x p(x) ∧ ∃x q(x)
Theorem
∀x p(x) ∨ ∀x q(x) ⇒ ∀x [p(x) ∨ q(x)]
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Predicate Implications
Theorem
∀x p(x) ⇒ ∃x p(x)
Theorem
∃x [p(x) ∧ q(x)] ⇒ ∃x p(x) ∧ ∃x q(x)
Theorem
∀x p(x) ∨ ∀x q(x) ⇒ ∀x [p(x) ∨ q(x)]
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Multiple Quantifiers
∃x∃y p(x , y)
∀x∃y p(x , y)
∃x∀y p(x , y)
∀x∀y p(x , y)
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Multiple Quantifier Examples
Example
U = Zp(x , y) : x + y = 17
∀x∃y p(x , y):for every x there exists a y such that x + y = 17
∃y∀x p(x , y):there exists a y so that for all x , x + y = 17
what if U = N?
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Multiple Quantifier Examples
Example
U = Zp(x , y) : x + y = 17
∀x∃y p(x , y):for every x there exists a y such that x + y = 17
∃y∀x p(x , y):there exists a y so that for all x , x + y = 17
what if U = N?
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Multiple Quantifier Examples
Example
U = Zp(x , y) : x + y = 17
∀x∃y p(x , y):for every x there exists a y such that x + y = 17
∃y∀x p(x , y):there exists a y so that for all x , x + y = 17
what if U = N?
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Multiple Quantifier Examples
Example
U = Zp(x , y) : x + y = 17
∀x∃y p(x , y):for every x there exists a y such that x + y = 17
∃y∀x p(x , y):there exists a y so that for all x , x + y = 17
what if U = N?
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Multiple Quantifiers
Example
Ux = {1, 2} ∧ Uy = {A,B}
∃x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∨ [p(2,A) ∨ p(2,B)]
∃x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∨ [p(2,A) ∧ p(2,B)]
∀x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∧ [p(2,A) ∨ p(2,B)]
∀x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∧ [p(2,A) ∧ p(2,B)]
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Multiple Quantifiers
Example
Ux = {1, 2} ∧ Uy = {A,B}
∃x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∨ [p(2,A) ∨ p(2,B)]
∃x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∨ [p(2,A) ∧ p(2,B)]
∀x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∧ [p(2,A) ∨ p(2,B)]
∀x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∧ [p(2,A) ∧ p(2,B)]
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Multiple Quantifiers
Example
Ux = {1, 2} ∧ Uy = {A,B}
∃x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∨ [p(2,A) ∨ p(2,B)]
∃x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∨ [p(2,A) ∧ p(2,B)]
∀x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∧ [p(2,A) ∨ p(2,B)]
∀x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∧ [p(2,A) ∧ p(2,B)]
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Multiple Quantifiers
Example
Ux = {1, 2} ∧ Uy = {A,B}
∃x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∨ [p(2,A) ∨ p(2,B)]
∃x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∨ [p(2,A) ∧ p(2,B)]
∀x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∧ [p(2,A) ∨ p(2,B)]
∀x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∧ [p(2,A) ∧ p(2,B)]
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Multiple Quantifiers
Example
Ux = {1, 2} ∧ Uy = {A,B}
∃x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∨ [p(2,A) ∨ p(2,B)]
∃x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∨ [p(2,A) ∧ p(2,B)]
∀x∃y p(x , y) ≡ [p(1,A) ∨ p(1,B)] ∧ [p(2,A) ∨ p(2,B)]
∀x∀y p(x , y) ≡ [p(1,A) ∧ p(1,B)] ∧ [p(2,A) ∧ p(2,B)]
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References
Required Reading: Grimaldi
Chapter 2: Fundamentals of Logic
2.4. The Use of Quantifiers
Supplementary Reading: O’Donnell, Hall, Page
Chapter 7: Predicate Logic
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Set
Definition
set: a collection of elements that are
distinct
unordered
non-repeating
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Set Representation
explicit representationelements are listed within braces: {a1, a2, . . . , an}
implicit representationelements that validate a predicate: {x |x ∈ G , p(x)}
∅: empty set
let S be a set, and a be an element
a ∈ S : a is an element of set Sa /∈ S : a is not an element of set S
|S |: number of elements (cardinality)
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Set Representation
explicit representationelements are listed within braces: {a1, a2, . . . , an}
implicit representationelements that validate a predicate: {x |x ∈ G , p(x)}
∅: empty set
let S be a set, and a be an element
a ∈ S : a is an element of set Sa /∈ S : a is not an element of set S
|S |: number of elements (cardinality)
![Page 50: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/50.jpg)
Set Representation
explicit representationelements are listed within braces: {a1, a2, . . . , an}
implicit representationelements that validate a predicate: {x |x ∈ G , p(x)}
∅: empty set
let S be a set, and a be an element
a ∈ S : a is an element of set Sa /∈ S : a is not an element of set S
|S |: number of elements (cardinality)
![Page 51: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/51.jpg)
Set Representation
explicit representationelements are listed within braces: {a1, a2, . . . , an}
implicit representationelements that validate a predicate: {x |x ∈ G , p(x)}
∅: empty set
let S be a set, and a be an element
a ∈ S : a is an element of set Sa /∈ S : a is not an element of set S
|S |: number of elements (cardinality)
![Page 52: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/52.jpg)
Set Representation
explicit representationelements are listed within braces: {a1, a2, . . . , an}
implicit representationelements that validate a predicate: {x |x ∈ G , p(x)}
∅: empty set
let S be a set, and a be an element
a ∈ S : a is an element of set Sa /∈ S : a is not an element of set S
|S |: number of elements (cardinality)
![Page 53: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/53.jpg)
Explicit Representation Example
Example
{3, 8, 2, 11, 5}11 ∈ {3, 8, 2, 11, 5}|{3, 8, 2, 11, 5}| = 5
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Implicit Representation Examples
Example
{x |x ∈ Z+, 20 < x3 < 100} ≡ {3, 4}{2x − 1|x ∈ Z+, 20 < x3 < 100} ≡ {5, 7}
Example
A = {x |x ∈ R, 1 ≤ x ≤ 5}
Example
E = {n|n ∈ N,∃k ∈ N [n = 2k]}A = {x |x ∈ E , 1 ≤ x ≤ 5}
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Implicit Representation Examples
Example
{x |x ∈ Z+, 20 < x3 < 100} ≡ {3, 4}{2x − 1|x ∈ Z+, 20 < x3 < 100} ≡ {5, 7}
Example
A = {x |x ∈ R, 1 ≤ x ≤ 5}
Example
E = {n|n ∈ N,∃k ∈ N [n = 2k]}A = {x |x ∈ E , 1 ≤ x ≤ 5}
![Page 56: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/56.jpg)
Implicit Representation Examples
Example
{x |x ∈ Z+, 20 < x3 < 100} ≡ {3, 4}{2x − 1|x ∈ Z+, 20 < x3 < 100} ≡ {5, 7}
Example
A = {x |x ∈ R, 1 ≤ x ≤ 5}
Example
E = {n|n ∈ N,∃k ∈ N [n = 2k]}A = {x |x ∈ E , 1 ≤ x ≤ 5}
![Page 57: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/57.jpg)
Set Dilemma
There is a barber who lives in a small town.He shaves all those men who don’t shave themselves.He doesn’t shave those men who shave themselves.
Does the barber shave himself?
yes → but he doesn’t shave men who shave themselves→ no
no → but he shaves all men who don’t shave themselves→ yes
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Set Dilemma
There is a barber who lives in a small town.He shaves all those men who don’t shave themselves.He doesn’t shave those men who shave themselves.
Does the barber shave himself?
yes → but he doesn’t shave men who shave themselves→ no
no → but he shaves all men who don’t shave themselves→ yes
![Page 59: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/59.jpg)
Set Dilemma
There is a barber who lives in a small town.He shaves all those men who don’t shave themselves.He doesn’t shave those men who shave themselves.
Does the barber shave himself?
yes → but he doesn’t shave men who shave themselves→ no
no → but he shaves all men who don’t shave themselves→ yes
![Page 60: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/60.jpg)
Set Dilemma
let S be the set of sets that are not an element of themselvesS = {A|A /∈ A}
Is S an element of itself?
yes → but the predicate is not valid → no
no → but the predicate is valid → yes
![Page 61: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/61.jpg)
Set Dilemma
let S be the set of sets that are not an element of themselvesS = {A|A /∈ A}
Is S an element of itself?
yes → but the predicate is not valid → no
no → but the predicate is valid → yes
![Page 62: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/62.jpg)
Set Dilemma
let S be the set of sets that are not an element of themselvesS = {A|A /∈ A}
Is S an element of itself?
yes → but the predicate is not valid → no
no → but the predicate is valid → yes
![Page 63: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/63.jpg)
Set Dilemma
let S be the set of sets that are not an element of themselvesS = {A|A /∈ A}
Is S an element of itself?
yes → but the predicate is not valid → no
no → but the predicate is valid → yes
![Page 64: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/64.jpg)
Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Subset
Definition
A ⊆ B ⇔ ∀x [x ∈ A → x ∈ B]
set equality:A = B ⇔ (A ⊆ B) ∧ (B ⊆ A)
proper subset:A ⊂ B ⇔ (A ⊆ B) ∧ (A 6= B)
∀S [∅ ⊆ S ]
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Subset
Definition
A ⊆ B ⇔ ∀x [x ∈ A → x ∈ B]
set equality:A = B ⇔ (A ⊆ B) ∧ (B ⊆ A)
proper subset:A ⊂ B ⇔ (A ⊆ B) ∧ (A 6= B)
∀S [∅ ⊆ S ]
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Subset
Definition
A ⊆ B ⇔ ∀x [x ∈ A → x ∈ B]
set equality:A = B ⇔ (A ⊆ B) ∧ (B ⊆ A)
proper subset:A ⊂ B ⇔ (A ⊆ B) ∧ (A 6= B)
∀S [∅ ⊆ S ]
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Subset
Definition
A ⊆ B ⇔ ∀x [x ∈ A → x ∈ B]
set equality:A = B ⇔ (A ⊆ B) ∧ (B ⊆ A)
proper subset:A ⊂ B ⇔ (A ⊆ B) ∧ (A 6= B)
∀S [∅ ⊆ S ]
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Subset
not a subset
A * B ⇔ ¬∀x [x ∈ A → x ∈ B]
⇔ ∃x ¬[x ∈ A → x ∈ B]
⇔ ∃x ¬[¬(x ∈ A) ∨ (x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ ¬(x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ (x /∈ B)]
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Subset
not a subset
A * B ⇔ ¬∀x [x ∈ A → x ∈ B]
⇔ ∃x ¬[x ∈ A → x ∈ B]
⇔ ∃x ¬[¬(x ∈ A) ∨ (x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ ¬(x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ (x /∈ B)]
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Subset
not a subset
A * B ⇔ ¬∀x [x ∈ A → x ∈ B]
⇔ ∃x ¬[x ∈ A → x ∈ B]
⇔ ∃x ¬[¬(x ∈ A) ∨ (x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ ¬(x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ (x /∈ B)]
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Subset
not a subset
A * B ⇔ ¬∀x [x ∈ A → x ∈ B]
⇔ ∃x ¬[x ∈ A → x ∈ B]
⇔ ∃x ¬[¬(x ∈ A) ∨ (x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ ¬(x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ (x /∈ B)]
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Subset
not a subset
A * B ⇔ ¬∀x [x ∈ A → x ∈ B]
⇔ ∃x ¬[x ∈ A → x ∈ B]
⇔ ∃x ¬[¬(x ∈ A) ∨ (x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ ¬(x ∈ B)]
⇔ ∃x [(x ∈ A) ∧ (x /∈ B)]
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Power Set
Definition
power set: P(S)the set of all subsets of a set, including the empty setand the set itself
if a set has n elements, its power set has 2n elements
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Power Set
Definition
power set: P(S)the set of all subsets of a set, including the empty setand the set itself
if a set has n elements, its power set has 2n elements
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Example of Power Set
Example
P({1, 2, 3}) = {∅,{1}, {2}, {3},{1, 2}, {1, 3}, {2, 3},{1, 2, 3}
}
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
![Page 78: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/78.jpg)
Set Operations
complement
A = {x |x /∈ A}
intersection
A ∩ B = {x |(x ∈ A) ∧ (x ∈ B)}
if A ∩ B = ∅ then A and B are disjoint
union
A ∪ B = {x |(x ∈ A) ∨ (x ∈ B)}
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Set Operations
complement
A = {x |x /∈ A}
intersection
A ∩ B = {x |(x ∈ A) ∧ (x ∈ B)}
if A ∩ B = ∅ then A and B are disjoint
union
A ∪ B = {x |(x ∈ A) ∨ (x ∈ B)}
![Page 80: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/80.jpg)
Set Operations
complement
A = {x |x /∈ A}
intersection
A ∩ B = {x |(x ∈ A) ∧ (x ∈ B)}
if A ∩ B = ∅ then A and B are disjoint
union
A ∪ B = {x |(x ∈ A) ∨ (x ∈ B)}
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Set Operations
difference
A− B = {x |(x ∈ A) ∧ (x /∈ B)}
A− B = A ∩ B
symmetric difference:A4 B = {x |(x ∈ A ∪ B) ∧ (x /∈ A ∩ B)}
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Set Operations
difference
A− B = {x |(x ∈ A) ∧ (x /∈ B)}
A− B = A ∩ B
symmetric difference:A4 B = {x |(x ∈ A ∪ B) ∧ (x /∈ A ∩ B)}
![Page 83: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/83.jpg)
Set Operations
difference
A− B = {x |(x ∈ A) ∧ (x /∈ B)}
A− B = A ∩ B
symmetric difference:A4 B = {x |(x ∈ A ∪ B) ∧ (x /∈ A ∩ B)}
![Page 84: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/84.jpg)
Cartesian Product
Definition
Cartesian product:A× B = {(a, b)|a ∈ A, b ∈ B}
A× B × C × · · · × N = {(a, b, . . . , n)|a ∈ A, b ∈ B, . . . , n ∈ N}
|A× B × C × · · · × N| = |A| · |B| · |C | · · · |N|
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Cartesian Product
Definition
Cartesian product:A× B = {(a, b)|a ∈ A, b ∈ B}
A× B × C × · · · × N = {(a, b, . . . , n)|a ∈ A, b ∈ B, . . . , n ∈ N}
|A× B × C × · · · × N| = |A| · |B| · |C | · · · |N|
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Cartesian Product Example
Example
A = {a1.a2, a3, a4}B = {b1, b2, b3}
A× B = {(a1, b1), (a1, b2), (a1, b3),
(a2, b1), (a2, b2), (a2, b3),
(a3, b1), (a3, b2), (a3, b3),
(a4, b1), (a4, b2), (a4, b3)
}
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Equivalences
Double Complement
A = A
CommutativityA ∩ B = B ∩ A A ∪ B = B ∪ A
Associativity(A ∩ B) ∩ C = A ∩ (B ∩ C ) (A ∪ B) ∪ C = A ∪ (B ∪ C )
IdempotenceA ∩ A = A A ∪ A = A
InverseA ∩ A = ∅ A ∪ A = U
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Equivalences
Double Complement
A = A
CommutativityA ∩ B = B ∩ A A ∪ B = B ∪ A
Associativity(A ∩ B) ∩ C = A ∩ (B ∩ C ) (A ∪ B) ∪ C = A ∪ (B ∪ C )
IdempotenceA ∩ A = A A ∪ A = A
InverseA ∩ A = ∅ A ∪ A = U
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Equivalences
Double Complement
A = A
CommutativityA ∩ B = B ∩ A A ∪ B = B ∪ A
Associativity(A ∩ B) ∩ C = A ∩ (B ∩ C ) (A ∪ B) ∪ C = A ∪ (B ∪ C )
IdempotenceA ∩ A = A A ∪ A = A
InverseA ∩ A = ∅ A ∪ A = U
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Equivalences
Double Complement
A = A
CommutativityA ∩ B = B ∩ A A ∪ B = B ∪ A
Associativity(A ∩ B) ∩ C = A ∩ (B ∩ C ) (A ∪ B) ∪ C = A ∪ (B ∪ C )
IdempotenceA ∩ A = A A ∪ A = A
InverseA ∩ A = ∅ A ∪ A = U
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Equivalences
Double Complement
A = A
CommutativityA ∩ B = B ∩ A A ∪ B = B ∪ A
Associativity(A ∩ B) ∩ C = A ∩ (B ∩ C ) (A ∪ B) ∪ C = A ∪ (B ∪ C )
IdempotenceA ∩ A = A A ∪ A = A
InverseA ∩ A = ∅ A ∪ A = U
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Equivalences
IdentityA ∩ U = A A ∪ ∅ = A
DominationA ∩ ∅ = ∅ A ∪ U = U
DistributivityA ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
AbsorptionA ∩ (A ∪ B) = A A ∪ (A ∩ B) = A
DeMorgan’s LawsA ∩ B = A ∪ B A ∪ B = A ∩ B
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Equivalences
IdentityA ∩ U = A A ∪ ∅ = A
DominationA ∩ ∅ = ∅ A ∪ U = U
DistributivityA ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
AbsorptionA ∩ (A ∪ B) = A A ∪ (A ∩ B) = A
DeMorgan’s LawsA ∩ B = A ∪ B A ∪ B = A ∩ B
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Equivalences
IdentityA ∩ U = A A ∪ ∅ = A
DominationA ∩ ∅ = ∅ A ∪ U = U
DistributivityA ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
AbsorptionA ∩ (A ∪ B) = A A ∪ (A ∩ B) = A
DeMorgan’s LawsA ∩ B = A ∪ B A ∪ B = A ∩ B
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Equivalences
IdentityA ∩ U = A A ∪ ∅ = A
DominationA ∩ ∅ = ∅ A ∪ U = U
DistributivityA ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
AbsorptionA ∩ (A ∪ B) = A A ∪ (A ∩ B) = A
DeMorgan’s LawsA ∩ B = A ∪ B A ∪ B = A ∩ B
![Page 96: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/96.jpg)
Equivalences
IdentityA ∩ U = A A ∪ ∅ = A
DominationA ∩ ∅ = ∅ A ∪ U = U
DistributivityA ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
AbsorptionA ∩ (A ∪ B) = A A ∪ (A ∩ B) = A
DeMorgan’s LawsA ∩ B = A ∪ B A ∪ B = A ∩ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
![Page 98: Discrete Mathematics - Predicates and Sets](https://reader034.vdocument.in/reader034/viewer/2022050919/5481c034b07959520c8b45f3/html5/thumbnails/98.jpg)
DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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DeMorgan’s Laws
Proof.
A ∩ B = {x |x /∈ (A ∩ B)}= {x |¬(x ∈ (A ∩ B))}= {x |¬((x ∈ A) ∧ (x ∈ B))}= {x |¬(x ∈ A) ∨ ¬(x ∈ B)}= {x |(x /∈ A) ∨ (x /∈ B)}= {x |(x ∈ A) ∨ (x ∈ B)}= {x |x ∈ A ∪ B}= A ∪ B
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Example of Equivalence
Theorem
A ∩ (B − C ) = (A ∩ B)− (A ∩ C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Equivalence Example
Proof.
(A ∩ B)− (A ∩ C ) = (A ∩ B) ∩ (A ∩ C )
= (A ∩ B) ∩ (A ∪ C )
= ((A ∩ B) ∩ A) ∪ ((A ∩ B) ∩ C ))
= ∅ ∪ ((A ∩ B) ∩ C ))
= (A ∩ B) ∩ C
= A ∩ (B ∩ C )
= A ∩ (B − C )
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Topics
1 PredicatesIntroductionQuantifiersMultiple Quantifiers
2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion
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Principle of Inclusion-Exclusion
|A ∪ B| = |A|+ |B| − |A ∩ B||A ∪ B ∪ C | =|A|+ |B|+ |C | − (|A ∩ B|+ |A ∩ C |+ |B ∩ C |) + |A ∩ B ∩ C |
Theorem
|A1 ∪ A2 ∪ · · · ∪ An| =∑
i
|Ai | −∑i ,j
|Ai ∩ Aj |
+∑i ,j ,k
|Ai ∩ Aj ∩ Ak |
· · ·+−1n−1|Ai ∩ Aj ∩ · · · ∩ An|
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Principle of Inclusion-Exclusion
|A ∪ B| = |A|+ |B| − |A ∩ B||A ∪ B ∪ C | =|A|+ |B|+ |C | − (|A ∩ B|+ |A ∩ C |+ |B ∩ C |) + |A ∩ B ∩ C |
Theorem
|A1 ∪ A2 ∪ · · · ∪ An| =∑
i
|Ai | −∑i ,j
|Ai ∩ Aj |
+∑i ,j ,k
|Ai ∩ Aj ∩ Ak |
· · ·+−1n−1|Ai ∩ Aj ∩ · · · ∩ An|
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Principle of Inclusion-Exclusion
|A ∪ B| = |A|+ |B| − |A ∩ B||A ∪ B ∪ C | =|A|+ |B|+ |C | − (|A ∩ B|+ |A ∩ C |+ |B ∩ C |) + |A ∩ B ∩ C |
Theorem
|A1 ∪ A2 ∪ · · · ∪ An| =∑
i
|Ai | −∑i ,j
|Ai ∩ Aj |
+∑i ,j ,k
|Ai ∩ Aj ∩ Ak |
· · ·+−1n−1|Ai ∩ Aj ∩ · · · ∩ An|
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
a method to identify prime numbers
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30
2 3 5 7 9 11 13 15 1719 21 23 25 27 29
2 3 5 7 11 13 1719 23 25 29
2 3 5 7 11 13 1719 23 29
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
a method to identify prime numbers
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30
2 3 5 7 9 11 13 15 1719 21 23 25 27 29
2 3 5 7 11 13 1719 23 25 29
2 3 5 7 11 13 1719 23 29
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
a method to identify prime numbers
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30
2 3 5 7 9 11 13 15 1719 21 23 25 27 29
2 3 5 7 11 13 1719 23 25 29
2 3 5 7 11 13 1719 23 29
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
a method to identify prime numbers
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30
2 3 5 7 9 11 13 15 1719 21 23 25 27 29
2 3 5 7 11 13 1719 23 25 29
2 3 5 7 11 13 1719 23 29
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
a method to identify prime numbers
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30
2 3 5 7 9 11 13 15 1719 21 23 25 27 29
2 3 5 7 11 13 1719 23 25 29
2 3 5 7 11 13 1719 23 29
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
number of primes between 1 and 100
numbers that are not divisible by 2, 3, 5 and 7
A2: set of numbers divisible by 2A3: set of numbers divisible by 3A5: set of numbers divisible by 5A7: set of numbers divisible by 7
|A2 ∪ A3 ∪ A5 ∪ A7|
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
number of primes between 1 and 100
numbers that are not divisible by 2, 3, 5 and 7
A2: set of numbers divisible by 2A3: set of numbers divisible by 3A5: set of numbers divisible by 5A7: set of numbers divisible by 7
|A2 ∪ A3 ∪ A5 ∪ A7|
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
number of primes between 1 and 100
numbers that are not divisible by 2, 3, 5 and 7
A2: set of numbers divisible by 2A3: set of numbers divisible by 3A5: set of numbers divisible by 5A7: set of numbers divisible by 7
|A2 ∪ A3 ∪ A5 ∪ A7|
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
|A2| = b100/2c = 50
|A3| = b100/3c = 33
|A5| = b100/5c = 20
|A7| = b100/7c = 14
|A2 ∩ A3| = b100/6c = 16
|A2 ∩ A5| = b100/10c = 10
|A2 ∩ A7| = b100/14c = 7
|A3 ∩ A5| = b100/15c = 6
|A3 ∩ A7| = b100/21c = 4
|A5 ∩ A7| = b100/35c = 2
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
|A2| = b100/2c = 50
|A3| = b100/3c = 33
|A5| = b100/5c = 20
|A7| = b100/7c = 14
|A2 ∩ A3| = b100/6c = 16
|A2 ∩ A5| = b100/10c = 10
|A2 ∩ A7| = b100/14c = 7
|A3 ∩ A5| = b100/15c = 6
|A3 ∩ A7| = b100/21c = 4
|A5 ∩ A7| = b100/35c = 2
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
|A2 ∩ A3 ∩ A5| = b100/30c = 3
|A2 ∩ A3 ∩ A7| = b100/42c = 2
|A2 ∩ A5 ∩ A7| = b100/70c = 1
|A3 ∩ A5 ∩ A7| = b100/105c = 0
|A2 ∩ A3 ∩ A5 ∩ A7| = b100/210c = 0
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
|A2 ∩ A3 ∩ A5| = b100/30c = 3
|A2 ∩ A3 ∩ A7| = b100/42c = 2
|A2 ∩ A5 ∩ A7| = b100/70c = 1
|A3 ∩ A5 ∩ A7| = b100/105c = 0
|A2 ∩ A3 ∩ A5 ∩ A7| = b100/210c = 0
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
|A2 ∪ A3 ∪ A5 ∪ A7| = (50 + 33 + 20 + 14)
− (16 + 10 + 7 + 6 + 4 + 2)
+ (3 + 2 + 1 + 0)
− (0)
= 78
number of primes: (100− 78) + 4− 1 = 25
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Inclusion-Exclusion Example
Example (sieve of Eratosthenes)
|A2 ∪ A3 ∪ A5 ∪ A7| = (50 + 33 + 20 + 14)
− (16 + 10 + 7 + 6 + 4 + 2)
+ (3 + 2 + 1 + 0)
− (0)
= 78
number of primes: (100− 78) + 4− 1 = 25
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References
Required Reading: Grimaldi
Chapter 3: Set Theory
3.1. Sets and Subsets3.2. Set Operations and the Laws of Set Theory
Chapter 8: The Principle of Inclusion and Exclusion
8.1. The Principle of Inclusion and Exclusion
Supplementary Reading: O’Donnell, Hall, Page
Chapter 8: Set Theory