Huan Long Shanghai Jiao Tong University Basic set theory
Relation
Function
Spring 2018 Georg Cantor(1845-1918) oGerman mathematician o Founder of set theory 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 contains𝑎𝑎 ∈ 𝐴𝐴. 𝑎𝑎 Important: 𝑎𝑎 ◦ Duplicates do not matter. ◦ Order does not matter.
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
V U A B U Spring 2018 { , , , } a {{a}} 𝑎𝑎 ∈ 𝑎𝑎 𝑒𝑒 𝑖𝑖 𝑜𝑜 𝑢𝑢 ∉ {{ }} ∅ ∉∅ {3,4,5}={5,4,3,4} ∅ ∈ ∅ ∈ ∅ S { } ∅⊆ S S ∅ ⊂ ∅ |{3, 3, 4, {2, 3},{1,2,{f}} }|=4 ⊆
Spring 2018 Union Intersection Difference Complement Symmetric difference Power set
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
A B UU
Spring 2018 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
A B
Spring 2018 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} 𝐴𝐴 − 𝐵𝐵 𝑥𝑥 𝑥𝑥 ∈ 𝐴𝐴 𝑏𝑏𝑏𝑏𝑏𝑏 𝑥𝑥∉𝐵𝐵 𝐴𝐴 ∩ 𝐵𝐵� Venn Diagram Representation
A B UU
Spring 2018 Definition Let U be the universal set. The complement of the sets A, denoted by or , is the complement of with respect to U. ̅ = 𝐴𝐴 −}𝐴𝐴= Example: -E = O 𝐴𝐴̅ 𝑥𝑥 𝑥𝑥∉𝐴𝐴 𝑈𝑈 − 𝐴𝐴 Venn Diagram Representation
A U
Spring 2018 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
A B ?
Spring 2018 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})={ɸ, {0},{1},{2}, {0,1},{0,2},{1,2},{0,1,2} } o P( )={ } P({ })={ ,{ }} o ∅ ∅ ∅ ∅ ∅ Spring 2018 1. Identity laws
2. Domination laws
3. Idempotent laws
Spring 2018 4. Complementation law
5. Commutative laws
6. Associative laws
Spring 2018 7. Distributive laws
8. De Morgan’s laws
Spring 2018
Spring 2018 Spring 2018 Basic set theory
Relation
Function
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
Spring 2018 A×B={
Spring 2018 Definition A relation is a set of ordered pairs. Examples o <={
ran ran R R
dom R
Spring 2018 A relation as a subset of the plane Let denote any binary relation on a set , we say: is reflexive, if ( )( ); 𝑅𝑅 𝑥𝑥 is symmetric, if ( , )( ); 𝑅𝑅 ∀𝑎𝑎 ∈ 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 is transitive , if , , [ ( )]; 𝑅𝑅 ∀𝑎𝑎 𝑏𝑏 ∈ 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 → 𝑏𝑏𝑏𝑏𝑏𝑏 𝑅𝑅 ∀𝑎𝑎 𝑏𝑏 𝑐𝑐 ∈ 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 ∧ 𝑏𝑏𝑏𝑏𝑏𝑏 → 𝑎𝑎𝑎𝑎𝑎𝑎
Spring 2018 Definition is an equivalence relation on iff is a binary relation on that is Reflexive𝑅𝑅 𝐴𝐴 𝑅𝑅o 𝐴𝐴 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 X R Y the set {[x]R |x ∈A} of all equivalence classes is a partition of A.
Spring 2018 X’ R Y’ Let = {0,1,2, … }; and is divisible by 6. Then is an equivalence relation on . The quotient set 𝜔𝜔 𝑚𝑚 ∼ 𝑛𝑛 ⇔ 𝑚𝑚 − 𝑛𝑛 has six members: 0 = ∼0,6,12, … , 𝜔𝜔 𝜔𝜔1⁄=∼ 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 Linear order/total order o transitive o trichotomy 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 Basic set theory
Relation
Function
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 (f∘g)(x)=f(g(x)) Inverse The inverse of F is the set ={
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 沈恩绍，集论与逻辑，科学出版社，（集合论部分）， 2001
Spring 2018 Huan Long Shanghai Jiao Tong University Paradox •Paradox and ZFC
Equinumerosity •Equinumerosity
Cardinal Numbers •Ordering
Infinite Cardinals •Countable sets
Spring 2018
Modern Axiomatic set theory set theory Paradox
Naive set theory
Spring 2018 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
Springmankind.… 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 Some properties from the first four natural numbers 0= 1={0}={ } ∅ 2={0,1}={ , { }} ∅ 3={0,1,2}={ , { }, { , { }}} ∅ ∅ ∅ ∅ ∅ ∅ 0∈ 1 ∈ 2 ∈ 3 ∈⋯ 0 1 2 3 ⋯
⊆ ⊆ ⊆ ⊆
Spring 2018 Paradox •Paradox and ZFC
Equinumerosity •Equinumerosity
Cardinal Numbers •Ordering
Infinite Cardinals •Countable sets
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 Example
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.
J(m,n)=((m+n)2+3m+n)/2
Spring 2018 Example: ω≈Q f: ω→ Q
Spring 2018 Example: (0,1) ≈ R (0,1)={x ∈ R | 0 f(x)= tan(π(2x-1)/2) Spring 2018 Examples (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 Spring 2018 Example: ℘(A) ≈ A2 For any set , we have 2. 𝐴𝐴 Proof: Define a 𝐴𝐴function from𝑃𝑃 𝐴𝐴( ≈) onto 2 as: For any subset of , ( ) is the characteristic𝐴𝐴 function of : 𝐻𝐻 𝑃𝑃 𝐴𝐴 1 𝐵𝐵 if 𝐴𝐴 𝐻𝐻 𝐵𝐵 = 𝐵𝐵 𝑥𝑥 ∈ 𝐵𝐵 𝐵𝐵 0 if 𝑓𝑓 is𝑥𝑥 one-to-one and onto. 𝑥𝑥 ∈ 𝐴𝐴 − 𝐵𝐵 𝐻𝐻 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 The set ω is not equinumerous to the set 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 Paradox •Paradox and ZFC Equinumerosity •Equinumerosity Cardinal Numbers •Ordering Infinite Cardinals •Countable sets Spring 2018 Ordering Cardinal Numbers Definition A set A is dominated by a set B (written A≼B) 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. A B iff A is equinumerous to some subset of B. ≼⊆ B F A B Spring 2018 Schröder-Bernstein Theorem If A B and B A, then A≈B. ≼ ≼ Spring 2018 Proof: : , : . Define by recursion: = and = 𝑓𝑓 𝐴𝐴 → 𝐵𝐵 𝑔𝑔 𝐵𝐵 → 𝐴𝐴 𝐶𝐶𝑛𝑛 = ( ) if + for some , 𝐶𝐶0 𝐴𝐴 − 𝑟𝑟𝑟𝑟𝑟𝑟 𝑔𝑔 𝐶𝐶𝑛𝑛 𝑔𝑔 𝑓𝑓 𝐶𝐶𝑛𝑛 g x otherwise 𝑛𝑛 ℎ 𝑥𝑥 𝑓𝑓−𝑥𝑥1 𝑥𝑥 ∈ 𝐶𝐶 𝑛𝑛 ( ) is one-to-one and onto. Spring 2018 ℎ 𝑥𝑥 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. ω A for any infinite A. ℵ0 ≼ ℵ0 ∙2 =? ℵ0 ℵ0 ℵ0 ℵ0 ℵ0 2 ≤ ℵ0 ∙2 ≤ 2 ∙2 =2 Spring 2018 Paradox •Paradox and ZFC Equinumerosity •Equinumerosity Cardinal Numbers •Ordering Infinite Cardinals •Countable sets Spring 2018 Countable Sets Definition A set A is countable iff A ω, 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 ℵ0 Are there any sets with cardinality between ℵ0 and 2 ? Continuum hypothesis (Cantor): No. ℵ0 i.e., there is no λ with ℵ0 < λ < 2 . 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 ⊢ ⊢