@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
Previous Lecture
Finishing the Logic Part
functional completeness
prenex normal form
inference rules in predicate logic
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 2 / 48
Today’s Topic
Set Theory (集合论)
naive set theory
axiomatic set theory
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 3 / 48
Textbooks
main textbook:
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7thedition [R]
auxiliary textbook:
Herbert B. Enderton, Elements of Set Theory [E]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 4 / 48
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
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
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
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
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
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
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
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))] .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 8 / 48
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))] .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 8 / 48
Naive Set Theory
Example
x = {a ∈ R | a2 − 3 · a + 2 = 0}y = {1, 2}
Then x = y by our equality principle.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 9 / 48
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
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
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
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
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
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))]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 12 / 48
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))]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 12 / 48
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
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
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))] .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 14 / 48
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))] .
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 14 / 48
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
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
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
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
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
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
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 17 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 17 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 17 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 17 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 17 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 18 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 18 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 18 / 48
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}}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 19 / 48
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}}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 19 / 48
Naive Set Theory
Problem
Is naive set theory enough?
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 20 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 21 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 21 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 21 / 48
Axiomatic Set Theory(公理化集合论)
[E], Page 17 – Page 33
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 22 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 23 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 24 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 24 / 48
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!
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 25 / 48
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!
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 25 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 26 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 26 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 27 / 48
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}.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 28 / 48
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}.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 29 / 48
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)).
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 30 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 31 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 31 / 48
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)}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 32 / 48
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)}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 32 / 48
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)}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 32 / 48
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)}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 32 / 48
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)}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 32 / 48
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)
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 33 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 34 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 34 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 34 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 34 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 34 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 35 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 35 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 35 / 48
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]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 35 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 36 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 36 / 48
Axiomatic Set Theory
Discussion
What is⋂∅?
Key Points
“vacuously truth”
“the set of all sets”
Answer⋂∅ is left undefined.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 37 / 48
Axiomatic Set Theory
Discussion
What is⋂∅?
Key Points
“vacuously truth”
“the set of all sets”
Answer⋂∅ is left undefined.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 37 / 48
Axiomatic Set Theory
Discussion
What is⋂∅?
Key Points
“vacuously truth”
“the set of all sets”
Answer⋂∅ is left undefined.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 37 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 38 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 38 / 48
Ordered Pairs and Cartesian Product
main textbook, Page 122 – Page 124auxiliary textbook, Page 35 – Page 38
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 39 / 48
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
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
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
Ordered Pairs
Tuples
(a, b, c) := ((a, b), c)
(a1, . . . , an) := ((a1, . . . , an−1), an)
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 41 / 48
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
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 42 / 48
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
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 42 / 48
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
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 42 / 48
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
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 42 / 48
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
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 42 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 43 / 48
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}
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 43 / 48
Summary
naive set theory
axiomatic set theory
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 44 / 48
Textbooks
main textbook:
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7thedition [R]
auxiliary textbook:
Herbert B. Enderton, Elements of Set Theory [E]
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 45 / 48
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
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 46 / 48
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.
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 47 / 48
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.)
Hongfei Fu (SJTU JHC) Set Theory Oct. 15th, 2019 48 / 48