Home >Documents >Huan Long Shanghai Jiao Tong University

# Huan Long Shanghai Jiao Tong University

Date post:22-Jan-2022
Category:
View:2 times
Transcript:
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
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
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
of 69/69
Huan Long Shanghai Jiao Tong University
Embed Size (px)
Recommended

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Documents

Education

Documents

Documents

Documents