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The Axiom of Unions Bernd Schr oder logo1 Bernd Schr oder - - PowerPoint PPT Presentation

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Unions Distributivity DeMorgans Laws Boolean Algebra The Axiom of Unions Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions Unions Distributivity DeMorgans

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SLIDE 1

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 2

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 3

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the family.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 4

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the

  • family. This set is denoted
  • C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-5
SLIDE 5

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the

  • family. This set is denoted
  • C and it is called the union of C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 6

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the

  • family. This set is denoted
  • C and it is called the union of C .

The union of two sets A,B is also denoted A∪B.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-7
SLIDE 7

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the

  • family. This set is denoted
  • C and it is called the union of C .

The union of two sets A,B is also denoted A∪B. Like equality in the previous presentation, unions cannot be defined based on the axioms we have so far.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-8
SLIDE 8

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the

  • family. This set is denoted
  • C and it is called the union of C .

The union of two sets A,B is also denoted A∪B. Like equality in the previous presentation, unions cannot be defined based on the axioms we have so far. That means we need this axiom, which is specifically geared towards unions.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 9

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

The Axiom of Unions

For every family C of sets, there exists a set whose elements are all the elements that belong to at least one element (set) of the

  • family. This set is denoted
  • C and it is called the union of C .

The union of two sets A,B is also denoted A∪B. Like equality in the previous presentation, unions cannot be defined based on the axioms we have so far. That means we need this axiom, which is specifically geared towards unions. The model at the end of the presentation for the Axiom of Specification satisfies the Axiom of Unions, too.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 11

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 12

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 13

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

A∪B

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 14

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

A∪B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-15
SLIDE 15

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A

A∪B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-16
SLIDE 16

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B B A

A∪B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-17
SLIDE 17

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B B A C

A∪B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 18

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B B A C

A∪B A∪B∪C

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 19

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B B A C

A∪B A∪B∪C

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 20

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proposition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 22

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 23

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .

Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 24

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .
  • Proof. We must prove that every element of C is an element of
  • C , too.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-25
SLIDE 25

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .
  • Proof. We must prove that every element of C is an element of
  • C , too. So let x ∈ C be arbitrary but fixed.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-26
SLIDE 26

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .
  • Proof. We must prove that every element of C is an element of
  • C , too. So let x ∈ C be arbitrary but fixed. By definition of

the union, x ∈

  • C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-27
SLIDE 27

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .
  • Proof. We must prove that every element of C is an element of
  • C , too. So let x ∈ C be arbitrary but fixed. By definition of

the union, x ∈

  • C . Because x ∈ C was arbitrary, every element
  • f C is in
  • C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-28
SLIDE 28

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .
  • Proof. We must prove that every element of C is an element of
  • C , too. So let x ∈ C be arbitrary but fixed. By definition of

the union, x ∈

  • C . Because x ∈ C was arbitrary, every element
  • f C is in
  • C . Thus C ⊆
  • C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-29
SLIDE 29

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Proposition. Let C be a family of sets and let C ∈ C . Then

C ⊆

  • C .
  • Proof. We must prove that every element of C is an element of
  • C , too. So let x ∈ C be arbitrary but fixed. By definition of

the union, x ∈

  • C . Because x ∈ C was arbitrary, every element
  • f C is in
  • C . Thus C ⊆
  • C .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-30
SLIDE 30

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

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SLIDE 31

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-32
SLIDE 32

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-33
SLIDE 33

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-34
SLIDE 34

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A. Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-35
SLIDE 35

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-36
SLIDE 36

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-37
SLIDE 37

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-38
SLIDE 38

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)

Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-39
SLIDE 39

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-40
SLIDE 40

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union. Let A,B,C be sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-41
SLIDE 41

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union. Let A,B,C be sets. Then

A∪(B∪C) = (A∪B)∪C.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-42
SLIDE 42

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union. Let A,B,C be sets. Then

A∪(B∪C) = (A∪B)∪C. Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-43
SLIDE 43

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union. Let A,B,C be sets. Then

A∪(B∪C) = (A∪B)∪C.

  • Proof. Simple exercise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-44
SLIDE 44

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union. Let A,B,C be sets. Then

A∪(B∪C) = (A∪B)∪C.

  • Proof. Simple exercise. (Maybe too simple.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-45
SLIDE 45

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Commutativity of set union. Let A,B be sets. Then

A∪B = B∪A.

  • Proof. Simple exercise. (Maybe too simple.)
  • Theorem. Associativity of set union. Let A,B,C be sets. Then

A∪(B∪C) = (A∪B)∪C.

  • Proof. Simple exercise. (Maybe too simple.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-46
SLIDE 46

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-47
SLIDE 47

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Distributivity of unions over intersections and vice

versa.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-48
SLIDE 48

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Distributivity of unions over intersections and vice
  • versa. Let A,B,C be sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-49
SLIDE 49

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Distributivity of unions over intersections and vice
  • versa. Let A,B,C be sets. Then the following hold.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-50
SLIDE 50

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Distributivity of unions over intersections and vice
  • versa. Let A,B,C be sets. Then the following hold.
  • 1. A∩(B∪C) = (A∩B)∪(A∩C).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-51
SLIDE 51

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Distributivity of unions over intersections and vice
  • versa. Let A,B,C be sets. Then the following hold.
  • 1. A∩(B∪C) = (A∩B)∪(A∩C).
  • 2. A∪(B∩C) = (A∪B)∩(A∪C).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-52
SLIDE 52

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-53
SLIDE 53

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-54
SLIDE 54

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

A∩B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-55
SLIDE 55

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A∩B

  • ∪

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-56
SLIDE 56

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A∩B

  • A∩C
  • ∪

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-57
SLIDE 57

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B A B C C

A∩B

  • A∩C
  • ∪

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-58
SLIDE 58

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B A B C C

A∩B A∩(B∪C)

  • A∩C
  • ∪

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-59
SLIDE 59

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-60
SLIDE 60

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

A∪B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-61
SLIDE 61

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A∪B

  • ∩

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-62
SLIDE 62

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A∪B A∪C

  • ∩

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-63
SLIDE 63

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B B A C C

A∪B A∪C

  • ∩

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-64
SLIDE 64

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B B A C C

A∪B A∪(B∩C) A∪C

  • ∩

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-65
SLIDE 65

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-66
SLIDE 66

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-67
SLIDE 67

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-68
SLIDE 68

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-69
SLIDE 69

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∨x ∈ C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-70
SLIDE 70

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : (x ∈ A∧x ∈ B)∨(x ∈ A∧x ∈ C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-71
SLIDE 71

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : (x ∈ A∧x ∈ B)∨(x ∈ A∧x ∈ C)
  • =
  • x ∈ A : (x ∈ A∩B)∨(x ∈ A∩C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-72
SLIDE 72

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : (x ∈ A∧x ∈ B)∨(x ∈ A∧x ∈ C)
  • =
  • x ∈ A : (x ∈ A∩B)∨(x ∈ A∩C)
  • =
  • x ∈ A : x ∈ (A∩B)∪(A∩C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-73
SLIDE 73

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : (x ∈ A∧x ∈ B)∨(x ∈ A∧x ∈ C)
  • =
  • x ∈ A : (x ∈ A∩B)∨(x ∈ A∩C)
  • =
  • x ∈ A : x ∈ (A∩B)∪(A∩C)
  • =

(A∩B)∪(A∩C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-74
SLIDE 74

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A∩(B∪C) =

  • x ∈ A : x ∈ A∩(B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : (x ∈ A∧x ∈ B)∨(x ∈ A∧x ∈ C)
  • =
  • x ∈ A : (x ∈ A∩B)∨(x ∈ A∩C)
  • =
  • x ∈ A : x ∈ (A∩B)∪(A∩C)
  • =

(A∩B)∪(A∩C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-75
SLIDE 75

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-76
SLIDE 76

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. DeMorgan’s Laws. Let A,B,C be sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-77
SLIDE 77

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. DeMorgan’s Laws. Let A,B,C be sets. Then the

following hold.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-78
SLIDE 78

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. DeMorgan’s Laws. Let A,B,C be sets. Then the

following hold.

  • 1. A\(B∪C) = (A\B)∩(A\C).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-79
SLIDE 79

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. DeMorgan’s Laws. Let A,B,C be sets. Then the

following hold.

  • 1. A\(B∪C) = (A\B)∩(A\C).
  • 2. A\(B∩C) = (A\B)∪(A\C).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-80
SLIDE 80

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-81
SLIDE 81

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-82
SLIDE 82

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

A\B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-83
SLIDE 83

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A\B ∩

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-84
SLIDE 84

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A\B A\C ∩

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-85
SLIDE 85

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B A B C C

A\B A\C ∩ =

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-86
SLIDE 86

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B A B C C

A\B A\(B∪C) A\C ∩ =

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-87
SLIDE 87

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-88
SLIDE 88

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

A\B

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-89
SLIDE 89

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A\B ∪

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-90
SLIDE 90

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B A C

A\B A\C ∪

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-91
SLIDE 91

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B B A C C

A\B A\C ∪ =

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-92
SLIDE 92

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B B A C C

A\B A\(B∩C) A\C ∪ =

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-93
SLIDE 93

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-94
SLIDE 94

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-95
SLIDE 95

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-96
SLIDE 96

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-97
SLIDE 97

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-98
SLIDE 98

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧¬(x ∈ C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-99
SLIDE 99

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧x ∈ A∧¬(x ∈ C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-100
SLIDE 100

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧x ∈ A∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A\B∧x ∈ A\C
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-101
SLIDE 101

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧x ∈ A∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A\B∧x ∈ A\C
  • =
  • x ∈ A : x ∈ (A\B)∩(A\C)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-102
SLIDE 102

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧x ∈ A∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A\B∧x ∈ A\C
  • =
  • x ∈ A : x ∈ (A\B)∩(A\C)
  • =

(A\B)∩(A\C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-103
SLIDE 103

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Proof (part 1 only). A\(B∪C) =

  • x ∈ A : x ∈ A\(B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∪C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B∨x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A∧¬(x ∈ B)∧x ∈ A∧¬(x ∈ C)
  • =
  • x ∈ A : x ∈ A\B∧x ∈ A\C
  • =
  • x ∈ A : x ∈ (A\B)∩(A\C)
  • =

(A\B)∩(A\C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-104
SLIDE 104

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Theorem.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-105
SLIDE 105

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-106
SLIDE 106

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets. Then the following hold.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-107
SLIDE 107

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets. Then the following hold.
  • 1. (A∪B)\C = (A\C)∪(B\C).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-108
SLIDE 108

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets. Then the following hold.
  • 1. (A∪B)\C = (A\C)∪(B\C).
  • 2. (A∩B)\C = (A\C)∩(B\C).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-109
SLIDE 109

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets. Then the following hold.
  • 1. (A∪B)\C = (A\C)∪(B\C).
  • 2. (A∩B)\C = (A\C)∩(B\C).

Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-110
SLIDE 110

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets. Then the following hold.
  • 1. (A∪B)\C = (A\C)∪(B\C).
  • 2. (A∩B)\C = (A\C)∩(B\C).
  • Proof. Good exercise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-111
SLIDE 111

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Theorem. Let A,B,C be sets. Then the following hold.
  • 1. (A∪B)\C = (A\C)∪(B\C).
  • 2. (A∩B)\C = (A\C)∩(B\C).
  • Proof. Good exercise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-112
SLIDE 112

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Example.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-113
SLIDE 113

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-114
SLIDE 114

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-115
SLIDE 115

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-116
SLIDE 116

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C) =

  • (A∪B)∪A
  • ∪C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-117
SLIDE 117

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C) =

  • (A∪B)∪A
  • ∪C

=

  • (B∪A)∪A
  • ∪C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-118
SLIDE 118

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C) =

  • (A∪B)∪A
  • ∪C

=

  • (B∪A)∪A
  • ∪C

=

  • B∪(A∪A)
  • ∪C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-119
SLIDE 119

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C) =

  • (A∪B)∪A
  • ∪C

=

  • (B∪A)∪A
  • ∪C

=

  • B∪(A∪A)
  • ∪C

= (B∪A)∪C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-120
SLIDE 120

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C) =

  • (A∪B)∪A
  • ∪C

=

  • (B∪A)∪A
  • ∪C

=

  • B∪(A∪A)
  • ∪C

= (B∪A)∪C = (A∪B)∪C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-121
SLIDE 121

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Example. Prove that (A∪B)∪(A∪C) = (A∪B)∪C.

Proof. (A∪B)∪(A∪C) =

  • (A∪B)∪A
  • ∪C

=

  • (B∪A)∪A
  • ∪C

=

  • B∪(A∪A)
  • ∪C

= (B∪A)∪C = (A∪B)∪C

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-122
SLIDE 122

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Definition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-123
SLIDE 123

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Definition. Let A,B be sets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-124
SLIDE 124

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

  • Definition. Let A,B be sets. Then the set

A∆B := (A\B)∪(B\A) is called the symmetric difference of A and B.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-125
SLIDE 125

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-126
SLIDE 126

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-127
SLIDE 127

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B

  • A\B

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-128
SLIDE 128

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B B A

  • A\B

∪

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-129
SLIDE 129

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A B B A

  • ❅

❅ ❅

  • ❅

❅ ❅ ❅

  • ❅

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅ A\B B\A ∪

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-130
SLIDE 130

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B B B A

  • ❅

❅ ❅

  • ❅

❅ ❅ ❅

  • ❅

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅ A\B B\A ∪ =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions

slide-131
SLIDE 131

logo1 Unions Distributivity DeMorgan’s Laws Boolean Algebra

Visualization

A A B B B A

  • ❅

❅ ❅ ❅ ❅ ❅

  • ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅

  • ❅

❅ ❅ ❅ ❅ ❅

  • ❅

❅ ❅ ❅

  • ❅

❅ ❅ ❅

  • ❅

❅ ❅ ❅ A\B B\A ∪ = A∆B

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Axiom of Unions