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@汥瑀瑯步渠 Set Theoryjhc.sjtu.edu.cn/~hongfeifu/lecture7.pdf · 2019-10-15 · Herbert B....
Transcript of @汥瑀瑯步渠 Set Theoryjhc.sjtu.edu.cn/~hongfeifu/lecture7.pdf · 2019-10-15 · Herbert B....
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Set Theory
Hongfei Fu
John Hopcroft Center for Computer ScienceShanghai Jiao Tong University
Oct. 15th, 2019
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 1 / 48
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Previous Lecture
Finishing the Logic Part
functional completeness
prenex normal form
inference rules in predicate logic
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Today’s Topic
Set Theory (集合论)
naive set theory
axiomatic set theory
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Textbooks
main textbook:
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7thedition [R]
auxiliary textbook:
Herbert B. Enderton, Elements of Set Theory [E]
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Naive Set Theory(朴素集合论)
main textbook, Page 115 – Page 125
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 5 / 48
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Naive Set Theory
What is a set?
A set is a collection of objects treated as a single entity.
Key Points
a collection of objects
a single entity (object)
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Naive Set Theory
What is a set?
A set is a collection of objects treated as a single entity.
Key Points
a collection of objects
a single entity (object)
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 6 / 48
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Naive Set Theory
What is a set?
A set is a collection of objects treated as a single entity.
Key Points
a collection of objects
a single entity (object)
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 6 / 48
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Naive Set Theory
Membership
a: an element/object
A: a set
We write that
a∈A if a is an element/member of A;
a 6∈A if a is not an element of A;
A Basic Principle
It should hold that either a∈A or a 6∈A but not both.
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Naive Set Theory
Membership
a: an element/object
A: a set
We write that
a∈A if a is an element/member of A;
a 6∈A if a is not an element of A;
A Basic Principle
It should hold that either a∈A or a 6∈A but not both.
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Naive Set Theory
Membership
a: an element/object
A: a set
We write that
a∈A if a is an element/member of A;
a 6∈A if a is not an element of A;
A Basic Principle
It should hold that either a∈A or a 6∈A but not both.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 7 / 48
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Naive Set Theory
Equalityx , y : sets
Two sets x and y are equal (i.e., they are the same set), written x = y , ifthey have the same members.
Logical Description
a: a variable whose domain is all objects
x , y : two variables whose domains are both all sets
Then we have that
∀x∀y [(x = y)↔∀a ((a∈ x)↔ (a∈ y))] .
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Naive Set Theory
Equalityx , y : sets
Two sets x and y are equal (i.e., they are the same set), written x = y , ifthey have the same members.
Logical Description
a: a variable whose domain is all objects
x , y : two variables whose domains are both all sets
Then we have that
∀x∀y [(x = y)↔∀a ((a∈ x)↔ (a∈ y))] .
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Naive Set Theory
Example
x = {a ∈ R | a2 − 3 · a + 2 = 0}y = {1, 2}
Then x = y by our equality principle.
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Naive Set Theory
Subsetsx , y : sets
Then we say that x is a subset of y , written x ⊆ y , if every element of x isan element of y .
Logical Description
a: a variable whose domain is all objects
x , y : two variables whose domains are both all sets
Then we have that
∀x∀y [(x ⊆ y)↔∀a ((a∈ x)→ (a∈ y))] .
Proper Subsets
x is a proper subset of y , written x ⊂ y , if x ⊆ y and x 6= y .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 10 / 48
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Naive Set Theory
Subsetsx , y : sets
Then we say that x is a subset of y , written x ⊆ y , if every element of x isan element of y .
Logical Description
a: a variable whose domain is all objects
x , y : two variables whose domains are both all sets
Then we have that
∀x∀y [(x ⊆ y)↔∀a ((a∈ x)→ (a∈ y))] .
Proper Subsets
x is a proper subset of y , written x ⊂ y , if x ⊆ y and x 6= y .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 10 / 48
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Naive Set Theory
Subsetsx , y : sets
Then we say that x is a subset of y , written x ⊆ y , if every element of x isan element of y .
Logical Description
a: a variable whose domain is all objects
x , y : two variables whose domains are both all sets
Then we have that
∀x∀y [(x ⊆ y)↔∀a ((a∈ x)→ (a∈ y))] .
Proper Subsets
x is a proper subset of y , written x ⊂ y , if x ⊆ y and x 6= y .
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Naive Set Theory
Examples
{2, 3}⊆{1, 2, 3, 5}{0, 1}⊆{x ∈ R | x2 − 2 · x ≤ 0}N⊆Q
An Important Property
x , y : two variables whose domains are both all sets
We have that
∀x∀y [(x = y)↔ ((x ⊆ y) ∧ (y ⊆ x))] .
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Naive Set Theory
Examples
{2, 3}⊆{1, 2, 3, 5}{0, 1}⊆{x ∈ R | x2 − 2 · x ≤ 0}N⊆Q
An Important Property
x , y : two variables whose domains are both all sets
We have that
∀x∀y [(x = y)↔ ((x ⊆ y) ∧ (y ⊆ x))] .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 11 / 48
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Naive Set Theory
Set Union (有限情况下的并集)x , y : sets
Then the union of sets x and y , written as x ∪ y , is the set consisting ofthe members of x together with the members of y .
Logical Description
a: a variable whose domain is all objects
x , y : two sets
Then we have that
∀a [(a∈ x ∪ y)↔ ((a∈ x)∨ (a∈ y))]
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Naive Set Theory
Set Union (有限情况下的并集)x , y : sets
Then the union of sets x and y , written as x ∪ y , is the set consisting ofthe members of x together with the members of y .
Logical Description
a: a variable whose domain is all objects
x , y : two sets
Then we have that
∀a [(a∈ x ∪ y)↔ ((a∈ x)∨ (a∈ y))]
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Naive Set Theory
Set Intersection (有限情况下的交集)x , y : sets
Then the intersection of sets x and y , written as x ∩ y , is the setconsisting of those objects that are members of both x and y .
Logical Description
a: a variable whose domain is all objects
x , y : two sets
Then we have that
∀a [(a∈ x ∩ y)↔ ((a∈ x)∧ (a∈ y))] .
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Naive Set Theory
Set Intersection (有限情况下的交集)x , y : sets
Then the intersection of sets x and y , written as x ∩ y , is the setconsisting of those objects that are members of both x and y .
Logical Description
a: a variable whose domain is all objects
x , y : two sets
Then we have that
∀a [(a∈ x ∩ y)↔ ((a∈ x)∧ (a∈ y))] .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 13 / 48
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Naive Set Theory
Set-Theoretic Difference (差集)x , y : sets
Then the (set-theoretic) difference of the set x w.r.t y , written as x − y orx \ y , is the set consisting of those elements of x that are not in y .
Logical Description
a: a variable whose domain is all objects
x , y : two sets
Then we have that
∀a [(a∈ x \ y)↔ ((a∈ x)∧ (a 6∈ y))] .
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Naive Set Theory
Set-Theoretic Difference (差集)x , y : sets
Then the (set-theoretic) difference of the set x w.r.t y , written as x − y orx \ y , is the set consisting of those elements of x that are not in y .
Logical Description
a: a variable whose domain is all objects
x , y : two sets
Then we have that
∀a [(a∈ x \ y)↔ ((a∈ x)∧ (a 6∈ y))] .
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Naive Set Theory
The Empty Set (空集)
The set ∅ is the set that contains no elements.
Logical Description
∀a (a 6∈ ∅) .
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Naive Set Theory
The Empty Set (空集)
The set ∅ is the set that contains no elements.
Logical Description
∀a (a 6∈ ∅) .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 15 / 48
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Naive Set Theory
Notations for Sets
a1, . . . , an: n objects
{a1, . . . , an}: the set consisting of exactly the elements a1, . . . , an
logical description: ∀a (a∈{a1, . . . , an} ↔∨n
i=1 a = ai )
Examples
{2, 3, 5}{1}{1, 2, 2}(= {1, 2})
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Naive Set Theory
Notations for Sets
a1, . . . , an: n objects
{a1, . . . , an}: the set consisting of exactly the elements a1, . . . , an
logical description: ∀a (a∈{a1, . . . , an} ↔∨n
i=1 a = ai )
Examples
{2, 3, 5}{1}{1, 2, 2}(= {1, 2})
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 16 / 48
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Naive Set Theory
Notations for Sets
a1, . . . , an: n objects
{a1, . . . , an}: the set consisting of exactly the elements a1, . . . , an
logical description: ∀a (a∈{a1, . . . , an} ↔∨n
i=1 a = ai )
Examples
{2, 3, 5}{1}{1, 2, 2}(= {1, 2})
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 16 / 48
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Naive Set Theory
Notations for Sets
a1, . . . , an: n objects
{a1, . . . , an}: the set consisting of exactly the elements a1, . . . , an
logical description: ∀a (a∈{a1, . . . , an} ↔∨n
i=1 a = ai )
Examples
{2, 3, 5}{1}{1, 2, 2}(= {1, 2})
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 16 / 48
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Naive Set Theory
Notations for Sets
P(a): a statement with the predicate P and the variable a
{a | P(a)}: the set consisting of exactly the elements b such thatP(b) holds
logical description: ∀b (b∈{a | P(a)} ↔ P(b))
Examples
[0, 1] = {a | a is a real number and 0 ≤ a ≤ 1}x ∪ y = {a | a ∈ x or a ∈ y}x ∩ y = {a | a ∈ x and a ∈ y}x \ y = {a | a ∈ x and a 6∈ y}
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Naive Set Theory
Notations for Sets
P(a): a statement with the predicate P and the variable a
{a | P(a)}: the set consisting of exactly the elements b such thatP(b) holds
logical description: ∀b (b∈{a | P(a)} ↔ P(b))
Examples
[0, 1] = {a | a is a real number and 0 ≤ a ≤ 1}x ∪ y = {a | a ∈ x or a ∈ y}x ∩ y = {a | a ∈ x and a ∈ y}x \ y = {a | a ∈ x and a 6∈ y}
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Naive Set Theory
Notations for Sets
P(a): a statement with the predicate P and the variable a
{a | P(a)}: the set consisting of exactly the elements b such thatP(b) holds
logical description: ∀b (b∈{a | P(a)} ↔ P(b))
Examples
[0, 1] = {a | a is a real number and 0 ≤ a ≤ 1}x ∪ y = {a | a ∈ x or a ∈ y}x ∩ y = {a | a ∈ x and a ∈ y}x \ y = {a | a ∈ x and a 6∈ y}
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Naive Set Theory
Notations for Sets
P(a): a statement with the predicate P and the variable a
{a | P(a)}: the set consisting of exactly the elements b such thatP(b) holds
logical description: ∀b (b∈{a | P(a)} ↔ P(b))
Examples
[0, 1] = {a | a is a real number and 0 ≤ a ≤ 1}
x ∪ y = {a | a ∈ x or a ∈ y}x ∩ y = {a | a ∈ x and a ∈ y}x \ y = {a | a ∈ x and a 6∈ y}
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Naive Set Theory
Notations for Sets
P(a): a statement with the predicate P and the variable a
{a | P(a)}: the set consisting of exactly the elements b such thatP(b) holds
logical description: ∀b (b∈{a | P(a)} ↔ P(b))
Examples
[0, 1] = {a | a is a real number and 0 ≤ a ≤ 1}x ∪ y = {a | a ∈ x or a ∈ y}x ∩ y = {a | a ∈ x and a ∈ y}x \ y = {a | a ∈ x and a 6∈ y}
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Naive Set Theory
Ordered Pairs (序对)
a, b: objects
(a, b): the ordered pair such that (a, b) = (c , d) iff a = c and b = d
Cartisian Product (笛卡尔积)x , y : sets
the Cartisian product: x × y := {(a, b) | a ∈ x ∧ b ∈ y}
Examples
R× RZ× Z{0, 1, 2, 3} × {100, 150, 200}
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Naive Set Theory
Ordered Pairs (序对)
a, b: objects
(a, b): the ordered pair such that (a, b) = (c , d) iff a = c and b = d
Cartisian Product (笛卡尔积)x , y : sets
the Cartisian product: x × y := {(a, b) | a ∈ x ∧ b ∈ y}
Examples
R× RZ× Z{0, 1, 2, 3} × {100, 150, 200}
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Naive Set Theory
Ordered Pairs (序对)
a, b: objects
(a, b): the ordered pair such that (a, b) = (c , d) iff a = c and b = d
Cartisian Product (笛卡尔积)x , y : sets
the Cartisian product: x × y := {(a, b) | a ∈ x ∧ b ∈ y}
Examples
R× RZ× Z{0, 1, 2, 3} × {100, 150, 200}
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Naive Set Theory
Power Set (幂集)x : a set
the power set 2x (or P(x)): the set consisting of all subsets of x
2x := {y | y ⊆ x}
Examples
2∅ = {∅}2{a,b} = {∅, {a}, {b}, {a, b}}
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Naive Set Theory
Power Set (幂集)x : a set
the power set 2x (or P(x)): the set consisting of all subsets of x
2x := {y | y ⊆ x}
Examples
2∅ = {∅}2{a,b} = {∅, {a}, {b}, {a, b}}
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Naive Set Theory
Problem
Is naive set theory enough?
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Naive Set Theory
Russell’s Paradox (罗素悖论)
X := {x | x 6∈ x};
the paradox:
X ∈ X implies X 6∈ X .X 6∈ X implies X ∈ X .
(optional) explanation:
The entity X conceptually exists, but is not a set.
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Naive Set Theory
Russell’s Paradox (罗素悖论)
X := {x | x 6∈ x};the paradox:
X ∈ X implies X 6∈ X .X 6∈ X implies X ∈ X .
(optional) explanation:
The entity X conceptually exists, but is not a set.
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Naive Set Theory
Russell’s Paradox (罗素悖论)
X := {x | x 6∈ x};the paradox:
X ∈ X implies X 6∈ X .X 6∈ X implies X ∈ X .
(optional) explanation:
The entity X conceptually exists, but is not a set.
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Axiomatic Set Theory(公理化集合论)
[E], Page 17 – Page 33
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Axiomatic Set Theory
The Principles
A set is a collection of objects treated as a single entity.
Every object is a set, and every set is an object.
It should hold that either a∈A or a 6∈A but not both.
A formal language is required for constructing meaningful statements.(will not be covered in the lecture)
Axioms are required for reasoning about sets.
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Axiomatic Set Theory
Axioms
Axioms are statements that are assumed to be true.
Why do we need axioms?
Axioms are basic rules for establishing correct statements.
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Axiomatic Set Theory
Axioms
Axioms are statements that are assumed to be true.
Why do we need axioms?
Axioms are basic rules for establishing correct statements.
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Axiomatic Set Theory
The Axiom of Extensionality (外延公理)
x , y , a: variables whose domains are sets
Then the axiom of extensionality says that
∀x∀y [(x = y)↔∀a ((a∈ x)↔ (a∈ y))] .
Impact
The only basic rule for judging whether two sets are equal!
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Axiomatic Set Theory
The Axiom of Extensionality (外延公理)
x , y , a: variables whose domains are sets
Then the axiom of extensionality says that
∀x∀y [(x = y)↔∀a ((a∈ x)↔ (a∈ y))] .
Impact
The only basic rule for judging whether two sets are equal!
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Axiomatic Set Theory
The Empty-Set Axiom (空集存在公理)
x , y : variables whose domains are sets
Then the empty-set axiom says that
∃x∀y(y 6∈ x)
where the set x is denoted by ∅.
Exercise
Prove through the axiom of extensionality that the empty set is unique.
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Axiomatic Set Theory
The Empty-Set Axiom (空集存在公理)
x , y : variables whose domains are sets
Then the empty-set axiom says that
∃x∀y(y 6∈ x)
where the set x is denoted by ∅.
Exercise
Prove through the axiom of extensionality that the empty set is unique.
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Axiomatic Set Theory
The Axiom of Set Union (有限情形下的并集公理)
x , y , z , a: variables whose domains are sets
Then the axiom of set union says that
∀x∀y∃z∀a [(a∈ z)↔ ((a∈ x)∨ (a∈ y))]
where such set z is denoted by x ∪ y .
Exercise
Prove through the axiom of extensionality that given any sets x , y , the setx ∪ y is unique.
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Axiomatic Set Theory
The Pairing Axiom (无序对集合存在公理)
x , y , z , a: variables whose domains are sets
Then the pairing axiom says that
∀x∀y∃z∀a [(a∈ z)↔ ((a= x)∨ (a= y))]
where such set z is denoted by {x , y}.
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Axiomatic Set Theory
Sets with Finitely Many Elements
a1, . . . , an: n objects
{a1, . . . , an}: the set consisting of exactly the elements a1, . . . , an
By the axioms of set unions and pairing, we can assert the existence of theset {a1, . . . , an}.
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Axiomatic Set Theory
The Power-Set Axiom (幂集公理)
x , y , z : variables whose domains are sets
Then the power-set axiom says that
∀x∃z∀y(y ∈ z ↔ y ⊆ x)
where the set z is denoted by 2x (or P(x)).
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Axiomatic Set Theory
The Subset Axioms (子集公理)
x , c : variables
φ(x): a predicate whose free variables are at most x
Then it is an axiom that
∀c ∃B ∀x (x ∈ B ↔ (x ∈ c ∧ φ(x)))
where B is denoted by {x ∈ c | φ(x)}.
The Role of c
The variable c represents the prescribed set over which the variable xranges over.
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Axiomatic Set Theory
The Subset Axioms (子集公理)
x , c : variables
φ(x): a predicate whose free variables are at most x
Then it is an axiom that
∀c ∃B ∀x (x ∈ B ↔ (x ∈ c ∧ φ(x)))
where B is denoted by {x ∈ c | φ(x)}.
The Role of c
The variable c represents the prescribed set over which the variable xranges over.
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Axiomatic Set Theory
Examples
{x ∈ R | x3 − 3 · x2 + 4 · x − 2 ≤ 0}x ∩ y := {a ∈ x ∪ y | a ∈ x ∧ a ∈ y}x \ y := {a ∈ x ∪ y | a ∈ x ∧ a 6∈ y}{2 · n | n ∈ Z} = {n ∈ Z | ∃k ∈ Z (n = 2 · k)}
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Axiomatic Set Theory
Examples
{x ∈ R | x3 − 3 · x2 + 4 · x − 2 ≤ 0}
x ∩ y := {a ∈ x ∪ y | a ∈ x ∧ a ∈ y}x \ y := {a ∈ x ∪ y | a ∈ x ∧ a 6∈ y}{2 · n | n ∈ Z} = {n ∈ Z | ∃k ∈ Z (n = 2 · k)}
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Axiomatic Set Theory
Examples
{x ∈ R | x3 − 3 · x2 + 4 · x − 2 ≤ 0}x ∩ y := {a ∈ x ∪ y | a ∈ x ∧ a ∈ y}x \ y := {a ∈ x ∪ y | a ∈ x ∧ a 6∈ y}
{2 · n | n ∈ Z} = {n ∈ Z | ∃k ∈ Z (n = 2 · k)}
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Axiomatic Set Theory
Examples
{x ∈ R | x3 − 3 · x2 + 4 · x − 2 ≤ 0}x ∩ y := {a ∈ x ∪ y | a ∈ x ∧ a ∈ y}x \ y := {a ∈ x ∪ y | a ∈ x ∧ a 6∈ y}{2 · n | n ∈ Z}
= {n ∈ Z | ∃k ∈ Z (n = 2 · k)}
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Axiomatic Set Theory
Examples
{x ∈ R | x3 − 3 · x2 + 4 · x − 2 ≤ 0}x ∩ y := {a ∈ x ∪ y | a ∈ x ∧ a ∈ y}x \ y := {a ∈ x ∪ y | a ∈ x ∧ a 6∈ y}{2 · n | n ∈ Z} = {n ∈ Z | ∃k ∈ Z (n = 2 · k)}
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Axiomatic Set Theory
Exercise
A,B,C : sets
Prove the De Morgan’s Law: C \ (A ∪ B) = (C \ A) ∩ (C \ B)
Prove the distributive Law: C ∩ (A ∪ B) = (C ∩ A) ∪ (C ∩ B)
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Axiomatic Set Theory
Axiom of Extended Union (广义并集公理)
The axiom of extended union says that
∀A∃B ∀a (a ∈ B ↔ (∃A (A ∈ A ∧ a ∈ A)))
where the set B is denoted by⋃A.
Example⋃{{n} | n ∈ Z} = Z⋃{[x − 1, x + 1] | x ∈ [0, 1]} = [−1, 2]
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Axiomatic Set Theory
Axiom of Extended Union (广义并集公理)
The axiom of extended union says that
∀A∃B ∀a (a ∈ B ↔ (∃A (A ∈ A ∧ a ∈ A)))
where the set B is denoted by⋃A.
Example⋃{{n} | n ∈ Z}
= Z⋃{[x − 1, x + 1] | x ∈ [0, 1]} = [−1, 2]
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Axiomatic Set Theory
Axiom of Extended Union (广义并集公理)
The axiom of extended union says that
∀A∃B ∀a (a ∈ B ↔ (∃A (A ∈ A ∧ a ∈ A)))
where the set B is denoted by⋃A.
Example⋃{{n} | n ∈ Z} = Z
⋃{[x − 1, x + 1] | x ∈ [0, 1]} = [−1, 2]
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Axiomatic Set Theory
Axiom of Extended Union (广义并集公理)
The axiom of extended union says that
∀A∃B ∀a (a ∈ B ↔ (∃A (A ∈ A ∧ a ∈ A)))
where the set B is denoted by⋃A.
Example⋃{{n} | n ∈ Z} = Z⋃{[x − 1, x + 1] | x ∈ [0, 1]}
= [−1, 2]
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Axiomatic Set Theory
Axiom of Extended Union (广义并集公理)
The axiom of extended union says that
∀A∃B ∀a (a ∈ B ↔ (∃A (A ∈ A ∧ a ∈ A)))
where the set B is denoted by⋃A.
Example⋃{{n} | n ∈ Z} = Z⋃{[x − 1, x + 1] | x ∈ [0, 1]} = [−1, 2]
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Axiomatic Set Theory
Extended Intersection (广义交集)
From the subset axiom, we also have extended intersection:
∀A(A 6= ∅ → ∃B ∀a (a ∈ B ↔ (∀A (A ∈ A → a ∈ A))))
where the set B is denoted by⋂A.
Proof
B = {a ∈ c | ∀A (A ∈ A → a ∈ A)} where c ∈ A .
Example⋂{[x − 1, x + 1] | x ∈ [0, 1]} = [0, 1]
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Axiomatic Set Theory
Extended Intersection (广义交集)
From the subset axiom, we also have extended intersection:
∀A(A 6= ∅ → ∃B ∀a (a ∈ B ↔ (∀A (A ∈ A → a ∈ A))))
where the set B is denoted by⋂A.
Proof
B = {a ∈ c | ∀A (A ∈ A → a ∈ A)} where c ∈ A .
Example⋂{[x − 1, x + 1] | x ∈ [0, 1]} = [0, 1]
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Axiomatic Set Theory
Extended Intersection (广义交集)
From the subset axiom, we also have extended intersection:
∀A(A 6= ∅ → ∃B ∀a (a ∈ B ↔ (∀A (A ∈ A → a ∈ A))))
where the set B is denoted by⋂A.
Proof
B = {a ∈ c | ∀A (A ∈ A → a ∈ A)} where c ∈ A .
Example⋂{[x − 1, x + 1] | x ∈ [0, 1]}
= [0, 1]
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Axiomatic Set Theory
Extended Intersection (广义交集)
From the subset axiom, we also have extended intersection:
∀A(A 6= ∅ → ∃B ∀a (a ∈ B ↔ (∀A (A ∈ A → a ∈ A))))
where the set B is denoted by⋂A.
Proof
B = {a ∈ c | ∀A (A ∈ A → a ∈ A)} where c ∈ A .
Example⋂{[x − 1, x + 1] | x ∈ [0, 1]} = [0, 1]
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Axiomatic Set Theory
Indexed Notation⋃n An =
⋃{A1, . . . ,An, . . . } =
⋃∞n=1 An⋂
n An =⋂{A1, . . . ,An, . . . } =
⋂∞n=1 An
⋃i∈I Ai =
⋃{Ai | i ∈ I} (I is an index set.)⋂
i∈I Ai =⋂{Ai | i ∈ I}
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Axiomatic Set Theory
Indexed Notation⋃n An =
⋃{A1, . . . ,An, . . . } =
⋃∞n=1 An⋂
n An =⋂{A1, . . . ,An, . . . } =
⋂∞n=1 An⋃
i∈I Ai =⋃{Ai | i ∈ I} (I is an index set.)⋂
i∈I Ai =⋂{Ai | i ∈ I}
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Axiomatic Set Theory
Discussion
What is⋂∅?
Key Points
“vacuously truth”
“the set of all sets”
Answer⋂∅ is left undefined.
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Axiomatic Set Theory
Discussion
What is⋂∅?
Key Points
“vacuously truth”
“the set of all sets”
Answer⋂∅ is left undefined.
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Axiomatic Set Theory
Discussion
What is⋂∅?
Key Points
“vacuously truth”
“the set of all sets”
Answer⋂∅ is left undefined.
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Axiomatic Set Theory
Theorem
There is no set to which every set belongs.
Proof
Suppose that there is such a set A. Then from the subset axiom, we candefine
B := {x ∈ A | x 6∈ x} .
Then we have that both B ∈ B and B 6∈ B holds. Contradiction.
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Axiomatic Set Theory
Theorem
There is no set to which every set belongs.
Proof
Suppose that there is such a set A. Then from the subset axiom, we candefine
B := {x ∈ A | x 6∈ x} .
Then we have that both B ∈ B and B 6∈ B holds. Contradiction.
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Ordered Pairs and Cartesian Product
main textbook, Page 122 – Page 124auxiliary textbook, Page 35 – Page 38
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Ordered Pairs
Ordered Pairs
example: coordinates (1, 2), (4.4, 7.3), . . .
key property:
(a, b) 6= (b, a) iff a 6= b(a, b) = (c , d) iff a = c and b = d
Definition
a, b: objects (sets)
Then we define that (a, b) := {{a}, {a, b}} .
Homework
Prove that (a, b) = (c , d) iff a = c and b = d .
Pay special attention to the case a = b.
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Ordered Pairs
Ordered Pairs
example: coordinates (1, 2), (4.4, 7.3), . . .
key property:
(a, b) 6= (b, a) iff a 6= b(a, b) = (c , d) iff a = c and b = d
Definition
a, b: objects (sets)
Then we define that (a, b) := {{a}, {a, b}} .
Homework
Prove that (a, b) = (c , d) iff a = c and b = d .
Pay special attention to the case a = b.
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Ordered Pairs
Ordered Pairs
example: coordinates (1, 2), (4.4, 7.3), . . .
key property:
(a, b) 6= (b, a) iff a 6= b(a, b) = (c , d) iff a = c and b = d
Definition
a, b: objects (sets)
Then we define that (a, b) := {{a}, {a, b}} .
Homework
Prove that (a, b) = (c , d) iff a = c and b = d .
Pay special attention to the case a = b.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 40 / 48
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Ordered Pairs
Tuples
(a, b, c) := ((a, b), c)
(a1, . . . , an) := ((a1, . . . , an−1), an)
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Cartesian Product
Definitionx , y : two sets
in naive set theory: x × y := {(a, b) | a ∈ x ∧ b ∈ y}
in axiomatic set theory: x × y := {(a, b) ∈ 22x ∪ y | a ∈ x ∧ b ∈ y}
The Reasoning
{a} ⊆ x ∪ y and {a, b} ⊆ x ∪ y(a, b) = {{a}, {a, b}} ⊆ 2x ∪ y
(a, b) ∈ 22x ∪ y
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Cartesian Product
Definitionx , y : two sets
in naive set theory: x × y := {(a, b) | a ∈ x ∧ b ∈ y}in axiomatic set theory: x × y := {(a, b) ∈ 22
x ∪ y | a ∈ x ∧ b ∈ y}
The Reasoning
{a} ⊆ x ∪ y and {a, b} ⊆ x ∪ y(a, b) = {{a}, {a, b}} ⊆ 2x ∪ y
(a, b) ∈ 22x ∪ y
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Cartesian Product
Definitionx , y : two sets
in naive set theory: x × y := {(a, b) | a ∈ x ∧ b ∈ y}in axiomatic set theory: x × y := {(a, b) ∈ 22
x ∪ y | a ∈ x ∧ b ∈ y}
The Reasoning
{a} ⊆ x ∪ y and {a, b} ⊆ x ∪ y
(a, b) = {{a}, {a, b}} ⊆ 2x ∪ y
(a, b) ∈ 22x ∪ y
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Cartesian Product
Definitionx , y : two sets
in naive set theory: x × y := {(a, b) | a ∈ x ∧ b ∈ y}in axiomatic set theory: x × y := {(a, b) ∈ 22
x ∪ y | a ∈ x ∧ b ∈ y}
The Reasoning
{a} ⊆ x ∪ y and {a, b} ⊆ x ∪ y(a, b) = {{a}, {a, b}} ⊆ 2x ∪ y
(a, b) ∈ 22x ∪ y
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Cartesian Product
Definitionx , y : two sets
in naive set theory: x × y := {(a, b) | a ∈ x ∧ b ∈ y}in axiomatic set theory: x × y := {(a, b) ∈ 22
x ∪ y | a ∈ x ∧ b ∈ y}
The Reasoning
{a} ⊆ x ∪ y and {a, b} ⊆ x ∪ y(a, b) = {{a}, {a, b}} ⊆ 2x ∪ y
(a, b) ∈ 22x ∪ y
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Cartisian Product
Cartisian Product of Multiple Sets
A× B × C := (A× B)× C
A1 × · · · × An := (A1 × · · · × An−1)× An
A1 × · · · × An = {(a1, . . . , an) | ak ∈ Ak for 1 ≤ k ≤ n}
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Cartisian Product
Cartisian Product of Multiple Sets
A× B × C := (A× B)× C
A1 × · · · × An := (A1 × · · · × An−1)× An
A1 × · · · × An = {(a1, . . . , an) | ak ∈ Ak for 1 ≤ k ≤ n}
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Summary
naive set theory
axiomatic set theory
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Textbooks
main textbook:
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7thedition [R]
auxiliary textbook:
Herbert B. Enderton, Elements of Set Theory [E]
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Reading
[R], Page 115 – Page 134
[E], Page 1 – Page 38
(optional) 石纯一等,数理逻辑与集合论(第二版),第九章
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Homeworks
[R], Page 125, Exercise 1(c), 10(c)(d)
[R], Page 126, Exercise 17, 18, 45
[R], Page 136, Exercise 24
[R], Page 137, Exercise 38(b), 40
[E], Page 26, Exercise 7(a)
[E], Page 38, Exercise 3
Note: In the homework, you can rely on Venn diagrams for intuition, butyou should write your homework using only formal proofs.
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Homeworks
Homework Submission
submission time: the start of the class on Oct. 22nd
teaching assistant:
Peixin Wang: [email protected] Wang: [email protected]
submission:
written version: submit on the desk (preferred)electronic version: word or pdf version, send email with title
“离散数学+姓名+学号+第六周周二”
to the teaching assistants (Students from the classes F1903001 –F1903004, please send to Peixin Wang. All other students please sendto Jinyi Wang.)
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