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Huan Long Shanghai Jiao Tong University

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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 .
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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 .
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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
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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
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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
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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
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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
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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
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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({∅})={∅,{∅}}
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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
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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}
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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 ∀, , ∈ [ ∧ → ()];
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Definition is an equivalence relation on iff is a binary relation on that is o Reflexive o Symmetric o Transitive
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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.
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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.
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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.
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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.
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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?)
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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.
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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.…
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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.
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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
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Natural Numbers in Set Theory
Constructing the natural numbers in terms of sets is part of the process of
“Embedding mathematics in set theory”
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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
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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}={∅, {∅}, {∅, {∅}}} ……
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0= ∅ 1={0}={∅} 2={0,1}={∅, {∅}} 3={0,1,2}={∅, {∅}, {∅, {∅}}}
0∈ 1 ∈ 2 ∈ 3 ∈ 0⊆1 ⊆ 2 ⊆ 3 ⊆
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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?
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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.
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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.
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f(x)= tan(π(2x-1)/2)
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(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)
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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.
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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:
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Theorem(Cantor 1873)
The set ω is not equinumerous to the set R of real numbers.
No set is equinumerous to its power set.
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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.
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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).
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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.
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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
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Schröder-Bernstein Theorem
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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].
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ℵ0 is the least infinite cardinal. i.e. ωA for
any infinite A. ℵ0 2
ℵ0 =? 2ℵ0≤ ℵ0 2
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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 .
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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.
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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 .
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4
Equivalence relation
John von Neumann
11
30
31
Countable Sets
of 69/69
Huan Long Shanghai Jiao Tong University
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