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Bertrand Russell(1872-1970) oBritish philosopher, logician,
mathematician, historian, and social critic.

Ernst Zermelo(1871-1953) oGerman mathematician, foundations of mathematics and hence on philosophy

David Hilbert (1862-1943) o German mathematicia, one of the most influential and universal mathematicians of the 19th and early 20th centuries.

Kurt Gödel(1906-1978) oAustrian American logician, mathematician, and philosopher. ZFC not ¬CH .

Paul Cohen(1934-2007) oAmerican mathematician, 1963: ZFC not CH,AC .

Spring 2018

Spring 2018

By Georg Cantor in 1870s:

A set is an unordered collection of objects. The objects are called the elements, or members, of the set. A set is

said to contain its elements.

Notation: ∈ Meaning that: is an element of the set A, or,

Set A contains .

Spring 2018

a∈A a is an element of the set A. a∉A a is NOT an element of the set A. Set of sets {{a,b},{1, 5.2}, k} ∅ the empty set, or the null set, is set that has no elements. A⊆B subset relation. Each element of A is also an element of B. A=B equal relation. A⊆B and B⊆A. A≠B A⊂B strict subset relation. If A⊆B and A≠B |A| cardinality of a set, or the number of distinct elements. Venn Diagram

Spring 2018 UV UBA

∈ {, , ,} a ∉{{a}} ∅ ∉∅ ∅ ∈ ∅ ∈ {{∅}} {3,4,5}={5,4,3,4} ∅⊆S ∅ ⊂{∅ } S ⊆S |{3, 3, 4, {2, 3},{1,2,{f}} }|=4

Spring 2018

Spring 2018

Definition Let A and B be sets. The union of the sets A and B, denoted by A∪B, is the set that contains those elements that are either in A or in B, or both.

A U B={x | x∈A or x∈B} Example: {1,3,5} U {1,2,3}={1,2,3,5} Venn Diagram representation

Spring 2018

A UUB

Definition Let A and B be sets. The intersection of the sets A and B, denoted by A ∩ B, is the set that containing those elements in both A and B.

A ∩ B={x | x∈A and x∈B} Example: {1,3,5} ∩ {1,2,3}={1,3} Venn Diagram Representation

Spring 2018

A B

Definition Let A and B be sets. The difference of the sets A and B, denoted by A - B, is the set that containing those elements in A but not in B.

− = ∈ ∉} = ∩ Example: {1,3,5}-{1,2,3}={5} Venn Diagram Representation

Spring 2018

A UUB

Definition Let U be the universal set. The complement of the sets A, denoted by or −, is the complement of with respect to U.

= ∉} = − Example: -E = O Venn Diagram Representation

Spring 2018

UA

Definition Let A and B be sets. The symmetric difference of A and B, denoted by A ⊕ B, is the set containing those elements in either A or B, but not in their intersection.

A ⊕ B={x| (x∈A ∨ x∈B) ∧ x∉ A∩B } =(A-B)∪(B-A)

Venn Diagram: A ⊕ B A ⊕ B ⊕ c

Spring 2018

A B ?

Many problems involves testing all combinations of elements of a set to see if they satisfy some property. To consider all such combinations of elements of a set S, we build a new set that has its members all the subsets of S.

Definition: Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S) or ℘S .

Example: o P({0,1,2})={, {0},{1},{2}, {0,1},{0,2},{1,2},{0,1,2} } o P(∅)={∅} o P({∅})={∅,{∅}}

Spring 2018

Spring 2018

Spring 2018

In set theory {1,2}={2,1} What if we need the object <1,2> that will

encode more information: o 1 is the first component o 2 is the second component

Generally, we say <x, y> =<u, v> iff x=u ∧ y=v

Spring 2018

A×B={<x,y> | x∈A ∧ y ∈B } is the Cartesian product of set A and set B.

Example A={1,2} B={a,b,c} A×B={<1,a>,<1,b>,<1,c>,

<2,a>,<2,b>,<2,c>}

Definition A relation is a set of ordered pairs. Examples

o <={<x,y>∈R×R| x is less than y} o M={<x,y> ∈People× People| x is married to y}

Spring 2018

A relation as a subset of the plane

Let denote any binary relation on a set , we say: is reflexive, if (∀ ∈ )(); is symmetric, if (∀, ∈ )( → ); is transitive , if ∀, , ∈ [ ∧ → ()];

Spring 2018

Definition is an equivalence relation on iff is a binary relation on that is o Reflexive o Symmetric o Transitive

Spring 2018

Definition A partition π of a set A is a set of nonempty subsets of A that is disjoint and exhaustive. i.e. (a) no two different sets in π have any

common elements, and (b) each element of A is in some set in π.

Spring 2018

If R is an equivalence relation on A, then the quotient set (equivalence class) A/R is defined as

A/R={ [x]R | ∈A } Where A/R is read as “A modulo R”

The natural map (or canonical map) α:A→A/R defined by

α(x)= [x]R

Theorem Assume that R is an equivalence relation on A. Then the set {[x]R |x ∈A} of all equivalence classes is a partition of A.

Spring 2018

X R Y

X’ R Y’

Let = {0,1,2, … }; and ∼ ⇔− is divisible by 6. Then ∼ is an equivalence relation on . The quotient set ⁄ ∼ has six members:

0 = 0,6,12, … , 1 = 1,7,13, … , …… 5 = 5,11,17, …

Clique (with self-circles on each node) : a graph in which every edge is presented. Take the existence of edge as a relation. Then the equivalence class decided by such relation over the graph would be clique.

Spring 2018

Partial order o reflexive o anti-symmetric o transitive

Well order o total order o every non-empty subset of S has a least element in this ordering.

Spring 2018

Spring 2018

Definition A function is a relation F such that for each x in dom F there is only one y such that x F y. And y is called the value of F at x.

Notation F(x)=y Example f(x) = x2 f : R → R, f(2) = 4, f(3) = 9, etc. Composition (fg)(x)=f(g(x)) Inverse The inverse of F is the set

−1={<u,v> | v F u} −1 is not necessarily a function (why?)

Spring 2018

We say that F is a function from A into B or that F maps A into B (written F: A→B) iff F is a function, dom F=A and ran F⊆B. o If, in addition, ran F=B, then F is a function from A onto

B. F is also named a surjective function. o If, in addition, for any x∈dom F, y∈dom F, with x≠y,

F(x)≠F(y), then F is an injective function. or one-to- one (or single-rooted).

o F is bijective function : f is surjective and injective.

Spring 2018

Main References o Herbert B. Enderton, Elements of Set Theory, ACADEMIC

PRESS, 1977 o Yiannis Moschovakis, Notes on Set Theory (Second

Edition), Springer, 2005 o Keith Devlin, The Joy of Sets: Fundamentals of

Contemporary Set Theory, Springer-Verlag, 1993 o Kenneth H. Rosen, Discrete Mathematics and Its

Applications (Sixth Edition), 2007 o

2001

•Paradox and ZFC Paradox

Russell`s paradox(1902) Bertrand Russell(1872-1970) British philosopher, logician, mathematician,

historian, and social critic. In 1950 Russell was awarded the Nobel Prize in

Literature, "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought."

What I have lived for? Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind.…

Spring 2018

Barber Paradox

Suppose there is a town with just one male barber. The barber shaves all and only those men in town who do not shave themselves.

Question: Does the barber shave himself? If the barber does NOT shave himself, then he MUST abide by the

rule and shave himself. If he DOES shave himself, according to the rule he will NOT shave

himself.

Spring 2018

Formal Proof

Theorem There is no set to which every set belongs. [Russell, 1902]

Proof: Let A be a set; we will construct a set not belonging to A. Let

B={x∈A | x∉x} We claim that B∉A. we have, by the construction of B.

B∈B iff B∈A and B∉B If B∈A, then this reduces to

B∈B iff B∉B, Which is impossible, since one side must be true and the other false. Hence B∉A

Spring 2018

Natural Numbers in Set Theory

Constructing the natural numbers in terms of sets is part of the process of

“Embedding mathematics in set theory”

Spring 2018

John von Neumann

December 28, 1903 – February 8, 1957. Hungarian American mathematician who made major contributions to a vast range of fields:

Logic and set theory Quantum mechanics Economics and game theory Mathematical statistics and econometrics Nuclear weapons Computer science

Spring 2018

Natural numbers

By von Neumann: Each natural number is the set of all smaller natural numbers.

0= ∅ 1={0}={∅} 2={0,1}={∅, {∅}} 3={0,1,2}={∅, {∅}, {∅, {∅}}} ……

Spring 2018

0= ∅ 1={0}={∅} 2={0,1}={∅, {∅}} 3={0,1,2}={∅, {∅}, {∅, {∅}}}

0∈ 1 ∈ 2 ∈ 3 ∈ 0⊆1 ⊆ 2 ⊆ 3 ⊆

Spring 2018

Motivation

To discuss the size of sets. Given two sets A and B, we want to consider such questions as: Do A and B have the same size? Does A have more elements than B?

Spring 2018

Equinumerosity

Definition A set A is equinumerous to a set B (written A≈B) iff there is a one-to-one function from A onto B.

A one-to-one function from A onto B is called a one- to-one correspondence between A and B.

Spring 2018

Example: ω× ω ≈ ω

The set ω × ω is equinumerous to ω. There is a function J mapping ω × ω one-to-one onto ω.

Spring 2018

f(x)= tan(π(2x-1)/2)

Spring 2018

(0,1) ≈ (n,m) Proof: f(x) = (n-m)x+m

(0,1) ≈ {x| x∈ω ∧ x>0} =(0,+∞) Proof: f(x)=1/x -1

[0,1] ≈ [0,1) Proof: f(x)=x if 0≤x<1 and x≠1/(2n), n∈ω

f(x)=1/(2n+1) if x=1/(2n), n∈ω [0,1) ≈ (0,1) Proof: f(x)=x if 0<x<1 and x≠1/(2n), n∈ω

f(0)=1/2 x=0 f(x)=1/(2n+1) if x=1/(2n), n∈ω

[0,1] ≈ (0,1)

Spring 2018

For any set , we have ≈ 2.

Proof: Define a function from () onto 2 as: For any subset of , () is the characteristic function of :

1 if ∈ =

0 if ∈ − is one-to-one and onto.

Spring 2018

Theorem

For any sets A, B and C: A ≈ A If A ≈ B then B ≈ A If A ≈ B and B ≈ C then A ≈ C. Proof:

Spring 2018

Theorem(Cantor 1873)

The set ω is not equinumerous to the set R of real numbers.

No set is equinumerous to its power set.

Spring 2018

R of real numbers.

Proof: show that for any functon f: ω→ R, there is a real number z not belonging to ran f

f(0) =32.4345…, f(1) =-43.334…, f(2) = 0.12418…,

…… z: the integer part is 0, and the (n+1)st decimal place of z is 7 unless the (n+1)st decimal place of f(n) is 7, in which case the (n+1)st decimal place of z is 6. Then z is a real number not in ran f.

Spring 2018

No set is equinumerous to its power set.

Proof: Let g: A→℘(A); we will construct a subset B of A that is not in ran g. Specifically, let

B={x∈ A | x∉ g(x)} Then B⊆A, but for each x∈ A

x∈ B iff x∉ g(x) Hence B≠g(x).

Spring 2018

Ordering Cardinal Numbers

Definition A set A is dominated by a set B (written AB) iff there is a one-to-one function from A into B.

Spring 2018

Examples

Any set dominates itself. If A⊆B, then A is dominated by B. AB iff A is equinumerous to some subset of B.

B F

A B

Spring 2018

Schröder-Bernstein Theorem

Spring 2018

Proof:

: → , : → . Define by recursion: 0 = − and + = = () if ∈ for some ,

g−1 x otherwise

Application of the Schröder- Bernstein Theorem

Example If A⊆B⊆C and A≈C, then all three sets are

equinumerous. The set R of real numbers is equinumerous

to the closed unit interval [0,1].

Spring 2018

ℵ0 is the least infinite cardinal. i.e. ωA for

any infinite A. ℵ0 2

ℵ0 =? 2ℵ0≤ ℵ0 2

Spring 2018

Definition A set A is countable iff Aω,

Intuitively speaking, the elements in a countable set can be counted by means of the natural numbers.

An equivalent definition: A set A is countable iff either A is finite or A ≈ ω .

Spring 2018

Example

ω is countable, as is Z and Q R is uncountable A, B are countable sets ∀ C⊆A, C is countable A∪B is countable A × B is countable

For any infinite set A, ℘(A) is uncountable.

Spring 2018

Continuum Hypothesis

Are there any sets with cardinality between ℵ0 and 2ℵ0 ? Continuum hypothesis (Cantor): No.

i.e., there is no λ with ℵ0 < λ < 2ℵ0 . Or, equivalently, it says: Every uncountable set of real numbers is

equinumerous to the set of all real numbers.

GENERAL VERSION: for any infinite cardinal κ, there is no cardinal number between κ and 2κ .

HISTORY Georg Cantor: 1878, proposed the conjecture David Hilbert: 1900, the first of Hilbert’s 23 problems. Kurt Gödel: 1939, ZFC ¬CH . Paul Cohen: 1963, ZFC CH .

Spring 2018

4

Equivalence relation

John von Neumann

11

30

31

Countable Sets

Ernst Zermelo(1871-1953) oGerman mathematician, foundations of mathematics and hence on philosophy

David Hilbert (1862-1943) o German mathematicia, one of the most influential and universal mathematicians of the 19th and early 20th centuries.

Kurt Gödel(1906-1978) oAustrian American logician, mathematician, and philosopher. ZFC not ¬CH .

Paul Cohen(1934-2007) oAmerican mathematician, 1963: ZFC not CH,AC .

Spring 2018

Spring 2018

By Georg Cantor in 1870s:

A set is an unordered collection of objects. The objects are called the elements, or members, of the set. A set is

said to contain its elements.

Notation: ∈ Meaning that: is an element of the set A, or,

Set A contains .

Spring 2018

a∈A a is an element of the set A. a∉A a is NOT an element of the set A. Set of sets {{a,b},{1, 5.2}, k} ∅ the empty set, or the null set, is set that has no elements. A⊆B subset relation. Each element of A is also an element of B. A=B equal relation. A⊆B and B⊆A. A≠B A⊂B strict subset relation. If A⊆B and A≠B |A| cardinality of a set, or the number of distinct elements. Venn Diagram

Spring 2018 UV UBA

∈ {, , ,} a ∉{{a}} ∅ ∉∅ ∅ ∈ ∅ ∈ {{∅}} {3,4,5}={5,4,3,4} ∅⊆S ∅ ⊂{∅ } S ⊆S |{3, 3, 4, {2, 3},{1,2,{f}} }|=4

Spring 2018

Spring 2018

Definition Let A and B be sets. The union of the sets A and B, denoted by A∪B, is the set that contains those elements that are either in A or in B, or both.

A U B={x | x∈A or x∈B} Example: {1,3,5} U {1,2,3}={1,2,3,5} Venn Diagram representation

Spring 2018

A UUB

Definition Let A and B be sets. The intersection of the sets A and B, denoted by A ∩ B, is the set that containing those elements in both A and B.

A ∩ B={x | x∈A and x∈B} Example: {1,3,5} ∩ {1,2,3}={1,3} Venn Diagram Representation

Spring 2018

A B

Definition Let A and B be sets. The difference of the sets A and B, denoted by A - B, is the set that containing those elements in A but not in B.

− = ∈ ∉} = ∩ Example: {1,3,5}-{1,2,3}={5} Venn Diagram Representation

Spring 2018

A UUB

Definition Let U be the universal set. The complement of the sets A, denoted by or −, is the complement of with respect to U.

= ∉} = − Example: -E = O Venn Diagram Representation

Spring 2018

UA

Definition Let A and B be sets. The symmetric difference of A and B, denoted by A ⊕ B, is the set containing those elements in either A or B, but not in their intersection.

A ⊕ B={x| (x∈A ∨ x∈B) ∧ x∉ A∩B } =(A-B)∪(B-A)

Venn Diagram: A ⊕ B A ⊕ B ⊕ c

Spring 2018

A B ?

Many problems involves testing all combinations of elements of a set to see if they satisfy some property. To consider all such combinations of elements of a set S, we build a new set that has its members all the subsets of S.

Definition: Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S) or ℘S .

Example: o P({0,1,2})={, {0},{1},{2}, {0,1},{0,2},{1,2},{0,1,2} } o P(∅)={∅} o P({∅})={∅,{∅}}

Spring 2018

Spring 2018

Spring 2018

In set theory {1,2}={2,1} What if we need the object <1,2> that will

encode more information: o 1 is the first component o 2 is the second component

Generally, we say <x, y> =<u, v> iff x=u ∧ y=v

Spring 2018

A×B={<x,y> | x∈A ∧ y ∈B } is the Cartesian product of set A and set B.

Example A={1,2} B={a,b,c} A×B={<1,a>,<1,b>,<1,c>,

<2,a>,<2,b>,<2,c>}

Definition A relation is a set of ordered pairs. Examples

o <={<x,y>∈R×R| x is less than y} o M={<x,y> ∈People× People| x is married to y}

Spring 2018

A relation as a subset of the plane

Let denote any binary relation on a set , we say: is reflexive, if (∀ ∈ )(); is symmetric, if (∀, ∈ )( → ); is transitive , if ∀, , ∈ [ ∧ → ()];

Spring 2018

Definition is an equivalence relation on iff is a binary relation on that is o Reflexive o Symmetric o Transitive

Spring 2018

Definition A partition π of a set A is a set of nonempty subsets of A that is disjoint and exhaustive. i.e. (a) no two different sets in π have any

common elements, and (b) each element of A is in some set in π.

Spring 2018

If R is an equivalence relation on A, then the quotient set (equivalence class) A/R is defined as

A/R={ [x]R | ∈A } Where A/R is read as “A modulo R”

The natural map (or canonical map) α:A→A/R defined by

α(x)= [x]R

Theorem Assume that R is an equivalence relation on A. Then the set {[x]R |x ∈A} of all equivalence classes is a partition of A.

Spring 2018

X R Y

X’ R Y’

Let = {0,1,2, … }; and ∼ ⇔− is divisible by 6. Then ∼ is an equivalence relation on . The quotient set ⁄ ∼ has six members:

0 = 0,6,12, … , 1 = 1,7,13, … , …… 5 = 5,11,17, …

Clique (with self-circles on each node) : a graph in which every edge is presented. Take the existence of edge as a relation. Then the equivalence class decided by such relation over the graph would be clique.

Spring 2018

Partial order o reflexive o anti-symmetric o transitive

Well order o total order o every non-empty subset of S has a least element in this ordering.

Spring 2018

Spring 2018

Definition A function is a relation F such that for each x in dom F there is only one y such that x F y. And y is called the value of F at x.

Notation F(x)=y Example f(x) = x2 f : R → R, f(2) = 4, f(3) = 9, etc. Composition (fg)(x)=f(g(x)) Inverse The inverse of F is the set

−1={<u,v> | v F u} −1 is not necessarily a function (why?)

Spring 2018

We say that F is a function from A into B or that F maps A into B (written F: A→B) iff F is a function, dom F=A and ran F⊆B. o If, in addition, ran F=B, then F is a function from A onto

B. F is also named a surjective function. o If, in addition, for any x∈dom F, y∈dom F, with x≠y,

F(x)≠F(y), then F is an injective function. or one-to- one (or single-rooted).

o F is bijective function : f is surjective and injective.

Spring 2018

Main References o Herbert B. Enderton, Elements of Set Theory, ACADEMIC

PRESS, 1977 o Yiannis Moschovakis, Notes on Set Theory (Second

Edition), Springer, 2005 o Keith Devlin, The Joy of Sets: Fundamentals of

Contemporary Set Theory, Springer-Verlag, 1993 o Kenneth H. Rosen, Discrete Mathematics and Its

Applications (Sixth Edition), 2007 o

2001

•Paradox and ZFC Paradox

Russell`s paradox(1902) Bertrand Russell(1872-1970) British philosopher, logician, mathematician,

historian, and social critic. In 1950 Russell was awarded the Nobel Prize in

Literature, "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought."

What I have lived for? Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind.…

Spring 2018

Barber Paradox

Suppose there is a town with just one male barber. The barber shaves all and only those men in town who do not shave themselves.

Question: Does the barber shave himself? If the barber does NOT shave himself, then he MUST abide by the

rule and shave himself. If he DOES shave himself, according to the rule he will NOT shave

himself.

Spring 2018

Formal Proof

Theorem There is no set to which every set belongs. [Russell, 1902]

Proof: Let A be a set; we will construct a set not belonging to A. Let

B={x∈A | x∉x} We claim that B∉A. we have, by the construction of B.

B∈B iff B∈A and B∉B If B∈A, then this reduces to

B∈B iff B∉B, Which is impossible, since one side must be true and the other false. Hence B∉A

Spring 2018

Natural Numbers in Set Theory

Constructing the natural numbers in terms of sets is part of the process of

“Embedding mathematics in set theory”

Spring 2018

John von Neumann

December 28, 1903 – February 8, 1957. Hungarian American mathematician who made major contributions to a vast range of fields:

Logic and set theory Quantum mechanics Economics and game theory Mathematical statistics and econometrics Nuclear weapons Computer science

Spring 2018

Natural numbers

By von Neumann: Each natural number is the set of all smaller natural numbers.

0= ∅ 1={0}={∅} 2={0,1}={∅, {∅}} 3={0,1,2}={∅, {∅}, {∅, {∅}}} ……

Spring 2018

0= ∅ 1={0}={∅} 2={0,1}={∅, {∅}} 3={0,1,2}={∅, {∅}, {∅, {∅}}}

0∈ 1 ∈ 2 ∈ 3 ∈ 0⊆1 ⊆ 2 ⊆ 3 ⊆

Spring 2018

Motivation

To discuss the size of sets. Given two sets A and B, we want to consider such questions as: Do A and B have the same size? Does A have more elements than B?

Spring 2018

Equinumerosity

Definition A set A is equinumerous to a set B (written A≈B) iff there is a one-to-one function from A onto B.

A one-to-one function from A onto B is called a one- to-one correspondence between A and B.

Spring 2018

Example: ω× ω ≈ ω

The set ω × ω is equinumerous to ω. There is a function J mapping ω × ω one-to-one onto ω.

Spring 2018

f(x)= tan(π(2x-1)/2)

Spring 2018

(0,1) ≈ (n,m) Proof: f(x) = (n-m)x+m

(0,1) ≈ {x| x∈ω ∧ x>0} =(0,+∞) Proof: f(x)=1/x -1

[0,1] ≈ [0,1) Proof: f(x)=x if 0≤x<1 and x≠1/(2n), n∈ω

f(x)=1/(2n+1) if x=1/(2n), n∈ω [0,1) ≈ (0,1) Proof: f(x)=x if 0<x<1 and x≠1/(2n), n∈ω

f(0)=1/2 x=0 f(x)=1/(2n+1) if x=1/(2n), n∈ω

[0,1] ≈ (0,1)

Spring 2018

For any set , we have ≈ 2.

Proof: Define a function from () onto 2 as: For any subset of , () is the characteristic function of :

1 if ∈ =

0 if ∈ − is one-to-one and onto.

Spring 2018

Theorem

For any sets A, B and C: A ≈ A If A ≈ B then B ≈ A If A ≈ B and B ≈ C then A ≈ C. Proof:

Spring 2018

Theorem(Cantor 1873)

The set ω is not equinumerous to the set R of real numbers.

No set is equinumerous to its power set.

Spring 2018

R of real numbers.

Proof: show that for any functon f: ω→ R, there is a real number z not belonging to ran f

f(0) =32.4345…, f(1) =-43.334…, f(2) = 0.12418…,

…… z: the integer part is 0, and the (n+1)st decimal place of z is 7 unless the (n+1)st decimal place of f(n) is 7, in which case the (n+1)st decimal place of z is 6. Then z is a real number not in ran f.

Spring 2018

No set is equinumerous to its power set.

Proof: Let g: A→℘(A); we will construct a subset B of A that is not in ran g. Specifically, let

B={x∈ A | x∉ g(x)} Then B⊆A, but for each x∈ A

x∈ B iff x∉ g(x) Hence B≠g(x).

Spring 2018

Ordering Cardinal Numbers

Definition A set A is dominated by a set B (written AB) iff there is a one-to-one function from A into B.

Spring 2018

Examples

Any set dominates itself. If A⊆B, then A is dominated by B. AB iff A is equinumerous to some subset of B.

B F

A B

Spring 2018

Schröder-Bernstein Theorem

Spring 2018

Proof:

: → , : → . Define by recursion: 0 = − and + = = () if ∈ for some ,

g−1 x otherwise

Application of the Schröder- Bernstein Theorem

Example If A⊆B⊆C and A≈C, then all three sets are

equinumerous. The set R of real numbers is equinumerous

to the closed unit interval [0,1].

Spring 2018

ℵ0 is the least infinite cardinal. i.e. ωA for

any infinite A. ℵ0 2

ℵ0 =? 2ℵ0≤ ℵ0 2

Spring 2018

Definition A set A is countable iff Aω,

Intuitively speaking, the elements in a countable set can be counted by means of the natural numbers.

An equivalent definition: A set A is countable iff either A is finite or A ≈ ω .

Spring 2018

Example

ω is countable, as is Z and Q R is uncountable A, B are countable sets ∀ C⊆A, C is countable A∪B is countable A × B is countable

For any infinite set A, ℘(A) is uncountable.

Spring 2018

Continuum Hypothesis

Are there any sets with cardinality between ℵ0 and 2ℵ0 ? Continuum hypothesis (Cantor): No.

i.e., there is no λ with ℵ0 < λ < 2ℵ0 . Or, equivalently, it says: Every uncountable set of real numbers is

equinumerous to the set of all real numbers.

GENERAL VERSION: for any infinite cardinal κ, there is no cardinal number between κ and 2κ .

HISTORY Georg Cantor: 1878, proposed the conjecture David Hilbert: 1900, the first of Hilbert’s 23 problems. Kurt Gödel: 1939, ZFC ¬CH . Paul Cohen: 1963, ZFC CH .

Spring 2018

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Equivalence relation

John von Neumann

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Countable Sets

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Huan Long Shanghai Jiao Tong University

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