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Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification Specifying Subsets

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logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

Bernd Schr¨

  • der

Bernd Schr¨

  • der

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

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SLIDE 2

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set, too.

Bernd Schr¨

  • der

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

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SLIDE 4

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set,

  • too. It is denoted
  • x ∈ S : p(x)
  • Bernd Schr¨
  • der

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

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SLIDE 5

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set,

  • too. It is denoted
  • x ∈ S : p(x)
  • r
  • x ∈ S | p(x)
  • .

Bernd Schr¨

  • der

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

slide-6
SLIDE 6

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set,

  • too. It is denoted
  • x ∈ S : p(x)
  • r
  • x ∈ S | p(x)
  • .

Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox.

Bernd Schr¨

  • der

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

slide-7
SLIDE 7

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set,

  • too. It is denoted
  • x ∈ S : p(x)
  • r
  • x ∈ S | p(x)
  • .

Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. The only difference is that the

  • bjects must come out of an existing set.

Bernd Schr¨

  • der

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

slide-8
SLIDE 8

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set,

  • too. It is denoted
  • x ∈ S : p(x)
  • r
  • x ∈ S | p(x)
  • .

Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. The only difference is that the

  • bjects must come out of an existing set. (But that’s crucial.)

Bernd Schr¨

  • der

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

slide-9
SLIDE 9

logo1 Specifying Subsets Empty Set Intersection Complement Model

The Axiom of Specification

If S is a set and p(·) is an open sentence for the elements of S, then the collection of all elements x ∈ S that satisfy p(x) is a set,

  • too. It is denoted
  • x ∈ S : p(x)
  • r
  • x ∈ S | p(x)
  • .

Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. The only difference is that the

  • bjects must come out of an existing set. (But that’s crucial.)

So now we have recaptured the parts of the assumptions in Russell’s Paradox that we definitely cannot live without.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.

Proof.

Bernd Schr¨

  • der

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

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SLIDE 12

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.
  • Proof. By our first axiom, there is a set S.

Bernd Schr¨

  • der

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

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SLIDE 13

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.
  • Proof. By our first axiom, there is a set S.

p(x) =

  • (x ∈ S)∧(x ∈ S)
  • is an open sentence.

Bernd Schr¨

  • der

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

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SLIDE 14

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.
  • Proof. By our first axiom, there is a set S.

p(x) =

  • (x ∈ S)∧(x ∈ S)
  • is an open sentence. By the Axiom of

Specification, A :=

  • x ∈ S : p(x)
  • =
  • x ∈ S : (x ∈ S)∧(x ∈ S)
  • is a set.

Bernd Schr¨

  • der

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

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SLIDE 15

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.
  • Proof. By our first axiom, there is a set S.

p(x) =

  • (x ∈ S)∧(x ∈ S)
  • is an open sentence. By the Axiom of

Specification, A :=

  • x ∈ S : p(x)
  • =
  • x ∈ S : (x ∈ S)∧(x ∈ S)
  • is a set. But because p(x) is always false, A has no elements.

Bernd Schr¨

  • der

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

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SLIDE 16

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.
  • Proof. By our first axiom, there is a set S.

p(x) =

  • (x ∈ S)∧(x ∈ S)
  • is an open sentence. By the Axiom of

Specification, A :=

  • x ∈ S : p(x)
  • =
  • x ∈ S : (x ∈ S)∧(x ∈ S)
  • is a set. But because p(x) is always false, A has no elements.

Bernd Schr¨

  • der

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

slide-17
SLIDE 17

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. There is a set that contains no elements.
  • Proof. By our first axiom, there is a set S.

p(x) =

  • (x ∈ S)∧(x ∈ S)
  • is an open sentence. By the Axiom of

Specification, A :=

  • x ∈ S : p(x)
  • =
  • x ∈ S : (x ∈ S)∧(x ∈ S)
  • is a set. But because p(x) is always false, A has no elements.

The set will be called the empty set, denoted / 0.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C. Proof.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C .

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a

set.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C .

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C .

Bernd Schr¨

  • der

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

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SLIDE 25

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C . Definition.

Bernd Schr¨

  • der

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

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SLIDE 26

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C .

  • Definition. Let C be a family of sets.

Bernd Schr¨

  • der

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

slide-27
SLIDE 27

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C .

  • Definition. Let C be a family of sets. The set of elements x so

that x ∈ C for all C ∈ C is called the intersection of C .

Bernd Schr¨

  • der

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

slide-28
SLIDE 28

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C .

  • Definition. Let C be a family of sets. The set of elements x so

that x ∈ C for all C ∈ C is called the intersection of C . The intersection of the family C is denoted

  • C .

Bernd Schr¨

  • der

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

slide-29
SLIDE 29

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a nonempty family of sets. Then there is

a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C.

  • Proof. ∀C ∈ C : x ∈ C is an open sentence. Let C0 ∈ C . By the

Axiom of Specification I :=

  • x ∈ C0 : [∀C ∈ C : x ∈ C]
  • is a
  • set. The elements of I are all the objects that are in all sets

C ∈ C .

  • Definition. Let C be a family of sets. The set of elements x so

that x ∈ C for all C ∈ C is called the intersection of C . The intersection of the family C is denoted

  • C . The intersection of

two sets A and B will also be denoted A∩B.

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

Bernd Schr¨

  • der

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

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logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 Bernd Schr¨

  • der

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

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SLIDE 32

logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 C2 Bernd Schr¨

  • der

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

slide-33
SLIDE 33

logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 C2

C1 ∩C2

  • Bernd Schr¨
  • der

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

slide-34
SLIDE 34

logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 C1 C2

C1 ∩C2

  • Bernd Schr¨
  • der

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

slide-35
SLIDE 35

logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 C1 C2 C2

C1 ∩C2

  • Bernd Schr¨
  • der

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

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SLIDE 36

logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 C1 C2 C2 C3

C1 ∩C2

  • Bernd Schr¨
  • der

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

slide-37
SLIDE 37

logo1 Specifying Subsets Empty Set Intersection Complement Model

Visualization With Venn Diagrams

C1 C1 C2 C2 C3

C1 ∩C2

{C1,C2,C3}

  • Bernd Schr¨
  • der

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

slide-38
SLIDE 38

logo1 Specifying Subsets Empty Set Intersection Complement Model

Proposition.

Bernd Schr¨

  • der

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

slide-39
SLIDE 39

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let C be a family of sets

Bernd Schr¨

  • der

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

slide-40
SLIDE 40

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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 Specification

slide-41
SLIDE 41

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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 Specification

slide-42
SLIDE 42

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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 Specification

slide-43
SLIDE 43

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too.

Bernd Schr¨

  • der

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

slide-44
SLIDE 44

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed.

Bernd Schr¨

  • der

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

slide-45
SLIDE 45

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C .

Bernd Schr¨

  • der

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

slide-46
SLIDE 46

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C . In particular, x ∈ C.

Bernd Schr¨

  • der

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

slide-47
SLIDE 47

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C . In particular, x ∈ C. Because x ∈

  • C was arbitrary, this means that
  • C ⊆ C.

Bernd Schr¨

  • der

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

slide-48
SLIDE 48

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C . In particular, x ∈ C. Because x ∈

  • C was arbitrary, this means that
  • C ⊆ C.

Bernd Schr¨

  • der

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

slide-49
SLIDE 49

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C . In particular, x ∈ C. Because x ∈

  • C was arbitrary, this means that
  • C ⊆ C.

Definition.

Bernd Schr¨

  • der

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

slide-50
SLIDE 50

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C . In particular, x ∈ C. Because x ∈

  • C was arbitrary, this means that
  • C ⊆ C.
  • Definition. Let C be a family of sets.

Bernd Schr¨

  • der

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

slide-51
SLIDE 51

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • 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
  • f C, too. Let x ∈
  • C be arbitrary but fixed. Then x is an

element of every set in C . In particular, x ∈ C. Because x ∈

  • C was arbitrary, this means that
  • C ⊆ C.
  • Definition. Let C be a family of sets. If
  • C = /

0, then the family C is called disjoint.

Bernd Schr¨

  • der

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

slide-52
SLIDE 52

logo1 Specifying Subsets Empty Set Intersection Complement Model

Proposition.

Bernd Schr¨

  • der

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

slide-53
SLIDE 53

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets.

Bernd Schr¨

  • der

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

slide-54
SLIDE 54

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B.

Bernd Schr¨

  • der

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

slide-55
SLIDE 55

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B. Proof.

Bernd Schr¨

  • der

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

slide-56
SLIDE 56

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B.

  • Proof. The desired set is {x ∈ A : x ∈ B}.

Bernd Schr¨

  • der

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

slide-57
SLIDE 57

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B.

  • Proof. The desired set is {x ∈ A : x ∈ B}.

Bernd Schr¨

  • der

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

slide-58
SLIDE 58

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B.

  • Proof. The desired set is {x ∈ A : x ∈ B}.
  • Definition. Let A and B be sets.

Bernd Schr¨

  • der

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

slide-59
SLIDE 59

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B.

  • Proof. The desired set is {x ∈ A : x ∈ B}.
  • Definition. Let A and B be sets. The set of elements of A that

are not in B is denoted A\B.

Bernd Schr¨

  • der

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

slide-60
SLIDE 60

logo1 Specifying Subsets Empty Set Intersection Complement Model

  • Proposition. Let A and B be sets. Then there is a set that

contains all elements of A that are not in B.

  • Proof. The desired set is {x ∈ A : x ∈ B}.
  • Definition. Let A and B be sets. The set of elements of A that

are not in B is denoted A\B. This set is called the (relative) complement of B in A.

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

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

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A A Bernd Schr¨

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

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A B B A Bernd Schr¨

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

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  • A\B

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

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A B B A

  • ❅

❅ ❅

  • ❅

❅ ❅ ❅

  • ❅

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅ A\B B\A

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

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A Model

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

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r r r

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

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/

r

/ Bernd Schr¨

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

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r r r r

{a} {a}

✒✑ ✓✏

/

r

/

❅ ❅ ❅

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

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r r r r r

{a} {a} {b} {b}

✒✑ ✓✏ ✒✑ ✓✏

/

r

/

❅ ❅ ❅

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

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r r r r r r

{a} {a} {b} {c} {b} {c}

✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏

/

r

/

❅ ❅ ❅

  • Bernd Schr¨
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Louisiana Tech University, College of Engineering and Science The Axiom of Specification

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r r r r r r r

{a} {a} {a,b} {b} {c} {b} {c}

✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏

/

✛ ✚ ✘ ✙

{a,b}

r

/

❅ ❅ ❅

  • ❅

❅ ❅

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

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r r r r r r r r

{a} {a} {a,b} {b} {a,c} {c} {b} {c}

✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏

/

✛ ✚ ✘ ✙

{a,b} {a,c}

r

/

❅ ❅ ❅

  • ❅

❅ ❅ ❅ ❅ ❅

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

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r r r r r r r r r

{a} {a} {a,b} {b} {a,c} {c} {b,c} {b} {c}

✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏

/

✛ ✚ ✘ ✙

{a,b} {a,c} {b,c}

r

/

❅ ❅ ❅

  • ❅

❅ ❅ ❅ ❅ ❅

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

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r r r r r r r r r

{a} {a} {a,b} {b} {a,c} {c} {b,c} {b} {c}

✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏

/

✛ ✚ ✘ ✙ ✬ ✫ ✩ ✪

{a,b} {a,c} {b,c} {a,b,c}

r r

/ {a,b,c}

❅ ❅ ❅

  • ❅

❅ ❅

  • ❅

❅ ❅ ❅ ❅ ❅

  • Bernd Schr¨
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Louisiana Tech University, College of Engineering and Science The Axiom of Specification