Axioms for Set Theory
|
|
- Gertrude Pitts
- 5 years ago
- Views:
Transcription
1 Axioms for Set Theory The following is a subset of the Zermelo-Fraenkel axioms for set theory. In this setting, all objects are sets which are denoted by letters, e.g. x, y, X, Y. Equality is logical identity: sets X, Y are equal (X = Y ) if and only if the letters X, Y denote the same set. There is a primitive notion of membership between sets (x X) which is read x is a member of X. 1. Equality Axiom: Sets X, Y are equal if they have the same members. ( X)( Y )(X = Y ( x)(x X x Y )) The set X is said to be a subset of Y every element of X is an element of Y. This statement is denoted by X Y. 2. Empty Set Axiom: There is a set with no elements. This set Z is denoted by {} or. ( Z)( x)x / Z 3. Doubleton Axiom: For any sets x, y there is a set whose elements are x, y. ( x)( y)( Z)(z Z (z = x z = y)) This set Z is denoted by {x, y}. The singleton set {x} is defined to be {x, x}. 4. Set Union Axiom: For any set X there is a set whose elements are the elements of the sets in the set X. ( X)( Z)(( z)(z Z ( Y )(Y X z Y ))) This set is denoted by X. If A, B are sets the union of A and B is defined to be {A, B}. This set is denoted by A B. We have x A B (x A x B). 5. Power Set Axiom: For any set X there is a set whose elements are the subsets of X. ( X)( Z)(z X z X) This set Z is called the power set of X and is denoted by P (X). 6. Set Formation Axiom: For any set A and any statement P (x) involving a variable set x there is a subset of A consisting of those elements x in A for which P (x) is true. ( X)( x)(x X (x A P (x))) This set is denoted by {x x A P (x)} or {x A P (x)}. This axiom is actually actually an axiom scheme; one axiom for each P (x). If C is a non-empty set, say C 1 C then ( C)(C C x C) x C 1 ( C)(C C = x C) so that the set C = {x ( C)(C C x C) exists by Axiom 6. The set C consists of those elements which lie in every set in the collection C. If C = {A, B} then C = {x x A x B} which is by definition A B, the intersection of the sets A and B. 1
2 7. Infinity Axiom: ( Z)( Z (z Z z {z} Z)) A set Z is called an inductive set if Z and z Z z {z} Z. The infinity axiom simply states that there is an inductive set. Theorem 1. If C is a non-empty collection of inductive sets then C is inductive. Proof. Let D = C. We have D since every set C C is inductive and hence C. Let x D. Then for every set C C we have x C which implies x {x} C since C is inductive. Thus x {x} D and D is inductive. Theorem 2. Let Z be an inductive set and let N be the intersection of the inductive subsets of Z. If D is any inductive set then N D. Proof. Since D Z is an inductive subset of Z we have N D Z and so N D. Definition 1. The set N of natural numbers is the smallest inductive set (with respect to the inclusion relation). The existence and uniqueness of N follows immediately from Theorem 2. If we define x + 1 to be x {x} and 0 =, 1 = = {0}, 2 = = {0, 1}, 3 = = {0, 1, 2}, 4 = = {0, 1, 2, 3},... then 0, 1, 2, 3, 4,... N. We now establish the elementary properties of the natural numbers. The first property which follows immediately from the fact that set N is the contained in every inductive set is the so-called Principle of Induction. If S is a subset of N which satisfies (a) 0 S (b) n S = n + 1 S then S = N. Proposition 1. ( n N)( m N)(m n = m n) Proof. Let S = {n N ( m N)(m n = m n)}. By the Principle of Induction is suffices to prove that S is inductive. We have 0 S since ( m N)(m 0 = m 0) is vacuously true as 0 = implies that m 0 is false for all m N. Now suppose that n S and that m n + 1. Then m n or m = n. If m n we have by our inductive hypothesis n S that m n which is also true if m = n. Since n n + 1 we obtain m n + 1 and hence that n + 1 S. Hence S is inductive. Corollary 1. ( m N)( n N)(m + 1 = n + 1 = m = n) Proof. If m + 1 = n + 1 then m n + 1 which implies m n or m = n and hence that m n by Proposition 1. Similarly n m + 1 implies n m. Hence m = n. Definition 2. The mapping σ : N N defined by σ(n) = n + 1 is called the successor mapping. We also denote σ(n) by n+. Theorem 3. The successor mapping is injective with image N {0}. 2
3 Proof. The injectivity follows from Corollary 1. The mapping σ is not surjective since 0 n + 1 for any n N. Now let S = {0} σ(n). It suffices to show that S = N. Since 0 we only have to show that n S = n + 1 S in order to prove that S is inductive. But n + 1 = σ(n) S for any n N so n + 1 S for any n S. Corollary 2. If n N and n 0 there is a unique m N with n = m + 1. This natural number m is denoted by n 1 or n and is called the immediate predecessor of n. Proposition 2. ( n N)n / n Proof. Let S = {n n / n}. Then 0 S since 0 / 0 =. Suppose that n S. Then n / n. We want to show that n + 1 / n + 1. Suppose to the contrary that n + 1 n + 1 = n {n}. Then n + 1 n or n + 1 = n which implies n + 1 n by Proposition 1. But then n n + 1 implies that n n which is a contradiction. Hence n + 1 S and S is inductive. Corollary 3. ( n N)( m N)(m n = m n) Proposition 3. ( n N)( m N)(m n m n) Proof. By Corollary 3 we only have to prove ( n N)( m N)(m n = m n. Let S = {n N ( m N)(m n = m n)}. Then 0 S since m 0 = is false for all m. Let n S and suppose that m n + 1 = n {n}. Then m n or n m. If m n then m = n or m n by the inductive hypothesis. In either case m n + 1. If n m then n m n + 1, which is not possible. Hence n + 1 S and S is inductive. Proposition 4. ( n N)( m N)(m n n m) Proof. By induction on n. Let S = {n N ( m N)(m n or n m)}. Then 0 S since 0 = m for any m. Let n S and let m N. Then m n or n m. But m n implies that m n + 1 and n m implies n m and hence n + 1 = n {n} m. Hence n + 1 S and S is inductive. Corollary 4. N is linearly ordered under inclusion. If m, n N we also denote m n by m n. Definition 3. For n N we let [0, n) denote the set {m N 0 m < n}. Note that by the definition of N we have [0, n) = n for any n N. Definition 4. Let X be a set. An infinite sequence of elements of X is a function with domain N and range a subset of X. If a is an infinite sequence it is denoted by (a n ) = (a 0, a 1,..., a n,...) where a n = a(n) = (n + 1) st term of a. A finite sequence of elements of X of length n is a function a with domain [0, n) and range a subset of X. If n = 0 the domain of the finite sequence is and the sequence is called the empty sequence. A finite sequence a of length n is usually denoted by (a 0, a 1,..., a n 1 ) with the convention that this denotes the empty sequence () when n = 0. An important method for constructing sequences is called recursion or definition by induction. The following theorem is an important special case called simple recursion. We will treat the general case later. 3
4 Theorem 3. Let X be a set, let x 0 X and let φ be a mapping of X into itself. Then there is a unique infinite sequence (a n ) such that a 0 = x 0 and a n+1 = φ(a n ) for n 0. Proof. We only give the main steps of the proof. The details of the proof are left to the reader. First one shows that for any n 1 there is a unique finite sequence a of length n such a 0 = x 0 and a m+1 = φ(a m ) for 0 m < n. Then, if a and b are two such finite sequences with a of length n and b of length p > n, one shows that a b, i.e. that a m = b m for m < n. If F is the collection of all such functions a of length 1 then F is the required infinite sequence. We now introduce the operations of addition and multiplication of natural numbers. For n N we define σ n : N N inductively by σ 0 = 1 N (the identity mapping on N), σ n+1 = σ σ n. We can similarly define f n for any mapping f : X X where X is any set. Proposition 4. ( n N)σ n (0) = n Proof. By induction on n. The proposition is true for n = 0 since σ 0 (0) = 1 N (0) = 0. If σ n (0) = n for some n N then σ n+1 (0) = σ(σ n (0)) = σ(n) = n + 1. Definition 3. Let m, n N. Then m + n = σ n (m) = σ n σ m (0), mn = (σ m ) n (0). We have m + 1 = σ(m), m + 0 = 0 + m = m, m 1 = 1 m = m, m 0 = 0 m = 0 and Theorem 4. Let m, n, p N. Then m + (n + 1) = (m + n) + 1, m(n + 1) = mn + m. (a) (m + n) + p = m + (n + p), (mn)p = m(np) (Associative Laws) (b) m + n = n + m, mn = nm (Commutative Laws) (c) m(n + p) = mn + mp, (m + n)p = mp + np (Distributive Laws) (d) m n ( p N)m = n + p (We denote p by m n.) Proof. (1) We first prove (m + n) + p = m + (n + p), the associative law for addition, by induction on p. Since (m + n) + 0 = m + n = m + (n + 0), it is true for p = 0. If it is true for some p then (m + n) + (p + 1) = ((m + n) + p) + 1 = (m + (n + p)) + 1 = m + ((n + p) + 1) = m + (n + (p + 1). Hence by induction it is true for all p. (2) We now prove m + n = n + m, the commutative law for addition, by induction. For n = 0 we have m + 0 = m = 0 + m. The proof of our inductive step will require the following auxiliary result. Lemma 1. ( n N)1 + n = n + 1 4
5 Proof of Lemma. We proceed by induction on n. For n = 0 we have = 0 = and if 1 + n = n + 1 for some n we have 1 + (n + 1) = (1 + n) + 1 = (n + 1) + 1. Resuming our inductive proof of the commutative law for addition, assume that m + n = n + m for some n. Then m + (n + 1) = (m + n) + 1 = (n + m) + 1 = 1 + (n + m) = (1 + n) + m = (n + 1) + m which completes the proof of the inductive step. (3) We now prove the distributive laws by induction on p. For p = 0 we have m(n + 0) = mn = mn + 0 = mn + m 0, (m + n) 0 = 0 = = m 0 + n 0. If we assume that m(n + p) = mn + mp and (m + n)p = mp + np for some p then m(n+(p+1)) = m((n+p)+1) = m(n+p)+1 = (mn+mp)+1 = mn+(mp+m) = mn+m(p+1) and (m + n)(p + 1) = (m + n)p + (m + n) = ((mp + np) + m) + n = (m + (mp + np)) + n = ((m + mp) + np) + n = (mp + m) + (np + n) = m(p + 1) + n(p + 1). (4) We leave the proofs of the associative and commutative laws for multiplication as well as (d) to the reader. Definition 5. A set X is said to be finite if there is a bijection f : [0, n) X for some n N and infinite otherwise. If f : [0, n) X and g : [0, n) X are bijections then h = g 1 f : [0, n) [0, m) is a bijection. The following theorem shows that m = n so that the natural number n in Definition 4 is uniquely determined by the finite set X. This number is called the cardinality of X and is denoted by X. Theorem 5. If h : [0, n) [0, m) is a bijection then n = m. Proof. By induction on n. If n = 0 then [0, n) = so [0, m) = and m = 0. Assume that for some natural number n h : [0, n) [0, m) bijective = n = m and let g : [0, n + 1) [0, m) be a bijection. Then m > 0 and we set p = m. Let q = g(n) and let f : [0, m) [0, m) be the bijection defined by f(p) = q, f(q) = p, and f(x) = x otherwise. Then h = f g : [0, n + 1) [0, m) is bijective and h(n) = p. It follows that the restriction of h to [0, n) is a bijection of [0, n) with [0, p). By our inductive hypothesis this gives n = p so that n + 1 = p + 1 = m. Theorem 6. If X is a finite set and Y X then Y is finite and Y X. If X = 0 then X = and so Y =, a finite set with Y = 0 = X. Suppose for some n N the statement is true for any finite set X with X = n. Let X be a finite set with X = n + 1, let 5
6 f : [0, n + 1) X be a bijection, let c = f(n) and let X = X {c}. Then g : [0, n) X defined by g(i) = f(i) is a bijection which shows that X is finite with X = n. Now Y = Y {c} X implies by the inductive hypothesis that Y finite with Y X. Since Y Y + 1 X + 1 = X we are done. Theorem 7. If X, Y are disjoint finite sets then X Y is finite and X Y = X + Y. Proof. Let f : [0, m) and g : [0, n) [0, n) be bijections. Then the mapping h : [0, m + n) X Y defined by { f(i) if 0 i < m, h(i) = g(i m) if m i < n is bijective since X Y =. Corollary 5. If X, Y are finite and f : X Y is injective then X Y with equality if and only if f is surjective. Proof. We have Y = f(x) + Y f(x) with f(x) = X. Corollary 6. N is an infinite set. Proof. σ : N N is injective but not surjective. Corollary 7. If X 1, X 2,..., X n are pairwise disjoint finite sets then X 1 X 1 X n is finite and X 1 X 1 X n = X 1 + X X n. Proof. The proof is by induction on n. The details are left to the reader. Note that n i=1 a i = a a n is defined inductively by 0 a i = 0 (the empty sum is zero), i=1 n+1 n a i = ( a i ) + a n+1. i=1 i=1 Theorem 8. If X, Y are finite then X Y, X Y are finite and X Y + X Y = X + Y. Proof. The set X Y is the union of the pairwise disjoint sets X X Y, Y X Y, X Y so that X Y = X X Y + Y X Y + X Y. Hence X Y + X Y = X X Y + X Y + Y X = X + Y. Theorem 9. If X, Y are finite then X Y is finite and X Y = X Y. Proof. The proof is by induction on Y. The inductive step follows from the fact that if c / Y then X (Y {c}) is the union of the disjoint sets X Y and X {c} and the fact that X {c} = X. The details are left to the reader. Theorem 10. A set X is infinite if and only if there is an injective mapping f : N X. Proof. The proof requires the following additional axiom 6
7 Axiom of Choice. If C is a collection of non-empty subsets of a set X then there is a function φ with domain C such that φ(c) C for any C C. We apply this axiom in the case C is the set of non-empty subsets of the infinite set X. We then define an injective function f : N X inductively by f(n) = φ(x f([0, n)). This is a stronger form of recursion in which the value of the function at n depends on the values f(m) for m < n. That such a function exists can be proven by a variant of the proof given for simple recursion. The details are left to the reader. Definition 6. An infinite set X is said to be countable if there is a bijection f : N X and uncountable otherwise. Theorem 11. P (N) is uncountable. Proof. P (N) is infinite since n {n} defines an injective mapping of N into P (N). If f is a mapping of N into P (N) let A be the set of those n N with n / f(n). If A = f(m) for some m N then m A implies m / f(m) by definition of A which implies m / A, a contradiction. On the other hand m / A implies m / f(m) which implies m A, again a contradiction. We are thus led to conclude that the hypothesis A = f(m) for some m is false which implies that f is not surjective. Hence there is no surjective mapping and hence no bijective mapping from N to X. Definition 7. If m, n N then m n N is defined inductively by m 0 = 1, m n+1 = m n m. Theorem 12. If X, Y are finite sets then the set X Y of all mappings of Y into X is finite of cardinality X Y. Proof. The proof is by induction on Y. The inductive step uses the fact that if c / Y the mapping of X Y {c} into X Y X, defined by f (f Y, f(c)), is bijective. Here f Y denotes the restriction of f to Y ; for y Y, we have f Y (y) = f(y). The details are left to the reader. Corollary 8. The number of finite sequences (a 0, a 1,..., a n 1 ) of length n where the terms a i lie in a finite set of cardinality m is m n. Corollary 9. P ([0, n)) = 2 n Proof. This follows from the fact that there is a bijection between P (X) and 2 X where 2 = {0, 1}. Theorem 13. Let X, Y be finite sets with X = m Y = n. Then the number of injections of X into Y is n(n 1)... (n m + 1). Proof. The proof is by induction on m. The inductive step uses the fact that if a / X, the set of injective mappings of X {a} into Y is in one-to-one correspondence with the set S of pairs (f, b) where f : X Y is injective and b Y f(x). One proves that if T is the set of injective mappings of X into Y then S = T (n m). To do this we partition S as follows. For each f T let S f be the set of those pairs (f, b) with b / f(x). Then S f = Y f(x) = (n m) which implies the result. Corollary 10. If X is a finite set then the number of bijections of X with itself is n(n 1) 1 = n!. 7
8 Corollary 11. Let X be a set with X = n. If C n,m denotes the number of Y subsets of X with Y = m then m!c m,n = n(n 1) (n m + 1). Proof. For any subset Y of X with Y = m the number of injective mappings f : [0, m) X with f([0, m)) = Y is m!. If m, n N with n = mk for some k N we say that m divides n in N and denote it by m n. If m 0 then k is unique and is denoted by n m or n/m. Thus the number of m element subsets of an n element set is n(n 1) (n m + 1) n! C n,m = = m! m!(n m)!. Theorem 14. Every non-empty set of natural numbers has a smallest element. i.e. the natural numbers are well-ordered. Proof.. Let S be a nonempty set of natural numbers and let m S. If m is the smallest element of S we are done otherwise there is an element p of S with p < m. Thus we are reduced to showing that the finite set S [0, m) has a smallest element. We leave to the reader of proving by induction that any finite non-empty set of natural numbers has a smallest element. Strong Induction. ( n N)(( m N)(m < n = P (m)) = P (n)) = ( n N)P (n) Proof. If ( n N)P (n) is false let n be the smallest natural number for which P (n) is false. Then ( m N)(m < n = P (m)) is true. But this implies P (n) true by hypothesis which is a contradiction. Hence P (n) is true for all n. It would seem that in the strong form of induction you don t have to prove P (0) is true. But you do since ( m N)(m < 0 = P (m)) = P (0) is true if and only iff P (0) is true. More generally, we have P (n) true for all n k if P (k) is true and for all n > k the truth of P (n) follows from the truth of P (m) for k m < n. The proof of this fact is left to the reader. 8
Chapter 1. Sets and Mappings
Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More informationADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS
ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationMATH 13 SAMPLE FINAL EXAM SOLUTIONS
MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More informationMath 4603: Advanced Calculus I, Summer 2016 University of Minnesota Notes on Cardinality of Sets
Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Notes on Cardinality of Sets Introduction In this short article, we will describe some basic notions on cardinality of sets. Given two
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More information6 CARDINALITY OF SETS
6 CARDINALITY OF SETS MATH10111 - Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationA BRIEF INTRODUCTION TO ZFC. Contents. 1. Motivation and Russel s Paradox
A BRIEF INTRODUCTION TO ZFC CHRISTOPHER WILSON Abstract. We present a basic axiomatic development of Zermelo-Fraenkel and Choice set theory, commonly abbreviated ZFC. This paper is aimed in particular
More informationHandout 2 (Correction of Handout 1 plus continued discussion/hw) Comments and Homework in Chapter 1
22M:132 Fall 07 J. Simon Handout 2 (Correction of Handout 1 plus continued discussion/hw) Comments and Homework in Chapter 1 Chapter 1 contains material on sets, functions, relations, and cardinality that
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationThe integers. Chapter 3
Chapter 3 The integers Recall that an abelian group is a set A with a special element 0, and operation + such that x +0=x x + y = y + x x +y + z) =x + y)+z every element x has an inverse x + y =0 We also
More informationMATH 3300 Test 1. Name: Student Id:
Name: Student Id: There are nine problems (check that you have 9 pages). Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Write your
More information1.4 Cardinality. Tom Lewis. Fall Term Tom Lewis () 1.4 Cardinality Fall Term / 9
1.4 Cardinality Tom Lewis Fall Term 2006 Tom Lewis () 1.4 Cardinality Fall Term 2006 1 / 9 Outline 1 Functions 2 Cardinality 3 Cantor s theorem Tom Lewis () 1.4 Cardinality Fall Term 2006 2 / 9 Functions
More informationReverse Mathematics of Topology
Reverse Mathematics of Topology William Chan 1 Abstract. This paper develops the Reverse Mathematics of second countable topologies, where the elements of the topological space exist. The notion of topology,
More informationShort notes on Axioms of set theory, Well orderings and Ordinal Numbers
Short notes on Axioms of set theory, Well orderings and Ordinal Numbers August 29, 2013 1 Logic and Notation Any formula in Mathematics can be stated using the symbols and the variables,,,, =, (, ) v j
More informationMath 455 Some notes on Cardinality and Transfinite Induction
Math 455 Some notes on Cardinality and Transfinite Induction (David Ross, UH-Manoa Dept. of Mathematics) 1 Cardinality Recall the following notions: function, relation, one-to-one, onto, on-to-one correspondence,
More informationNOTES ON WELL ORDERING AND ORDINAL NUMBERS. 1. Logic and Notation Any formula in Mathematics can be stated using the symbols
NOTES ON WELL ORDERING AND ORDINAL NUMBERS TH. SCHLUMPRECHT 1. Logic and Notation Any formula in Mathematics can be stated using the symbols,,,, =, (, ) and the variables v j : where j is a natural number.
More informationFunctions and cardinality (solutions) sections A and F TA: Clive Newstead 6 th May 2014
Functions and cardinality (solutions) 21-127 sections A and F TA: Clive Newstead 6 th May 2014 What follows is a somewhat hastily written collection of solutions for my review sheet. I have omitted some
More information20 Ordinals. Definition A set α is an ordinal iff: (i) α is transitive; and. (ii) α is linearly ordered by. Example 20.2.
20 Definition 20.1. A set α is an ordinal iff: (i) α is transitive; and (ii) α is linearly ordered by. Example 20.2. (a) Each natural number n is an ordinal. (b) ω is an ordinal. (a) ω {ω} is an ordinal.
More informationSection 2: Classes of Sets
Section 2: Classes of Sets Notation: If A, B are subsets of X, then A \ B denotes the set difference, A \ B = {x A : x B}. A B denotes the symmetric difference. A B = (A \ B) (B \ A) = (A B) \ (A B). Remarks
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationMAT115A-21 COMPLETE LECTURE NOTES
MAT115A-21 COMPLETE LECTURE NOTES NATHANIEL GALLUP 1. Introduction Number theory begins as the study of the natural numbers the integers N = {1, 2, 3,...}, Z = { 3, 2, 1, 0, 1, 2, 3,...}, and sometimes
More informationAN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationMath Fall 2014 Final Exam Solutions
Math 2001-003 Fall 2014 Final Exam Solutions Wednesday, December 17, 2014 Definition 1. The union of two sets X and Y is the set X Y consisting of all objects that are elements of X or of Y. The intersection
More informationInformal Statement Calculus
FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationWell Ordered Sets (continued)
Well Ordered Sets (continued) Theorem 8 Given any two well-ordered sets, either they are isomorphic, or one is isomorphic to an initial segment of the other. Proof Let a,< and b, be well-ordered sets.
More informationThis section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.
1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as
More informationNotes on ordinals and cardinals
Notes on ordinals and cardinals Reed Solomon 1 Background Terminology We will use the following notation for the common number systems: N = {0, 1, 2,...} = the natural numbers Z = {..., 2, 1, 0, 1, 2,...}
More informationFinite and Infinite Sets
Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following
More informationSolutions to Homework Set 1
Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement
More informationExercises for Unit VI (Infinite constructions in set theory)
Exercises for Unit VI (Infinite constructions in set theory) VI.1 : Indexed families and set theoretic operations (Halmos, 4, 8 9; Lipschutz, 5.3 5.4) Lipschutz : 5.3 5.6, 5.29 5.32, 9.14 1. Generalize
More informationJónsson posets and unary Jónsson algebras
Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal
More informationDO FIVE OUT OF SIX ON EACH SET PROBLEM SET
DO FIVE OUT OF SIX ON EACH SET PROBLEM SET 1. THE AXIOM OF FOUNDATION Early on in the book (page 6) it is indicated that throughout the formal development set is going to mean pure set, or set whose elements,
More informationSets, Structures, Numbers
Chapter 1 Sets, Structures, Numbers Abstract In this chapter we shall introduce most of the background needed to develop the foundations of mathematical analysis. We start with sets and algebraic structures.
More informationEconomics 204 Fall 2011 Problem Set 1 Suggested Solutions
Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.
More informationSETS AND FUNCTIONS JOSHUA BALLEW
SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationMeta-logic derivation rules
Meta-logic derivation rules Hans Halvorson February 19, 2013 Recall that the goal of this course is to learn how to prove things about (as opposed to by means of ) classical first-order logic. So, we will
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationS. Mrówka introduced a topological space ψ whose underlying set is the. natural numbers together with an infinite maximal almost disjoint family(madf)
PAYNE, CATHERINE ANN, M.A. On ψ (κ, M) spaces with κ = ω 1. (2010) Directed by Dr. Jerry Vaughan. 30pp. S. Mrówka introduced a topological space ψ whose underlying set is the natural numbers together with
More informationPeter Kahn. Spring 2007
MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 1 The Natural Numbers 1 1.1 The Peano Axioms............................ 2 1.2 Proof by induction............................ 4 1.3
More informationNumbers, sets, and functions
Chapter 1 Numbers, sets, and functions 1.1. The natural numbers, integers, and rational numbers We assume that you are familiar with the set of natural numbers the set of integers and the set of rational
More informationLecture Notes on Discrete Mathematics. October 15, 2018 DRAFT
Lecture Notes on Discrete Mathematics October 15, 2018 2 Contents 1 Basic Set Theory 5 1.1 Basic Set Theory....................................... 5 1.1.1 Union and Intersection of Sets...........................
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationREVIEW FOR THIRD 3200 MIDTERM
REVIEW FOR THIRD 3200 MIDTERM PETE L. CLARK 1) Show that for all integers n 2 we have 1 3 +... + (n 1) 3 < 1 n < 1 3 +... + n 3. Solution: We go by induction on n. Base Case (n = 2): We have (2 1) 3 =
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationPart II. Logic and Set Theory. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]
More informationIntroduction to Proofs
Introduction to Proofs Notes by Dr. Lynne H. Walling and Dr. Steffi Zegowitz September 018 The Introduction to Proofs course is organised into the following nine sections. 1. Introduction: sets and functions
More information3 COUNTABILITY AND CONNECTEDNESS AXIOMS
3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first
More informationCardinality and ordinal numbers
Cardinality and ordinal numbers The cardinality A of a finite set A is simply the number of elements in it. When it comes to infinite sets, we no longer can speak of the number of elements in such a set.
More informationS15 MA 274: Exam 3 Study Questions
S15 MA 274: Exam 3 Study Questions You can find solutions to some of these problems on the next page. These questions only pertain to material covered since Exam 2. The final exam is cumulative, so you
More information3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.
1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is
More informationPrinciples of Real Analysis I Fall I. The Real Number System
21-355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of real-valued functions of one real variable in a systematic and rigorous
More informationSection 7.5: Cardinality
Section 7: Cardinality In this section, we shall consider extend some of the ideas we have developed to infinite sets One interesting consequence of this discussion is that we shall see there are as many
More informationA Short Review of Cardinality
Christopher Heil A Short Review of Cardinality November 14, 2017 c 2017 Christopher Heil Chapter 1 Cardinality We will give a short review of the definition of cardinality and prove some facts about the
More informationChapter 2 Axiomatic Set Theory
Chapter 2 Axiomatic Set Theory Ernst Zermelo (1871 1953) was the first to find an axiomatization of set theory, and it was later expanded by Abraham Fraenkel (1891 1965). 2.1 Zermelo Fraenkel Set Theory
More informationPropositional Logic, Predicates, and Equivalence
Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationLet us first solve the midterm problem 4 before we bring up the related issues.
Math 310 Class Notes 6: Countability Let us first solve the midterm problem 4 before we bring up the related issues. Theorem 1. Let I n := {k N : k n}. Let f : I n N be a one-toone function and let Im(f)
More informationHomework 5. Solutions
Homework 5. Solutions 1. Let (X,T) be a topological space and let A,B be subsets of X. Show that the closure of their union is given by A B = A B. Since A B is a closed set that contains A B and A B is
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationSets are one of the basic building blocks for the types of objects considered in discrete mathematics.
Section 2.1 Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory
More informationCopyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction
Copyright & License Copyright c 2007 Jason Underdown Some rights reserved. statement sentential connectives negation conjunction disjunction implication or conditional antecedant & consequent hypothesis
More information2.1 Sets. Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R.
2. Basic Structures 2.1 Sets Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R. Definition 2 Objects in a set are called elements or members of the set. A set is
More informationFoundations of Mathematics
Foundations of Mathematics L. Pedro Poitevin 1. Preliminaries 1.1. Sets We will naively think of a set as a collection of mathematical objects, called its elements or members. To indicate that an object
More informationInitial Ordinals. Proposition 57 For every ordinal α there is an initial ordinal κ such that κ α and α κ.
Initial Ordinals We now return to ordinals in general and use them to give a more precise meaning to the notion of a cardinal. First we make some observations. Note that if there is an ordinal with a certain
More informationMath 3T03 - Topology
Math 3T03 - Topology Sang Woo Park April 5, 2018 Contents 1 Introduction to topology 2 1.1 What is topology?.......................... 2 1.2 Set theory............................... 3 2 Functions 4 3
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
More informationMATH 13 FINAL EXAM SOLUTIONS
MATH 13 FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers. T F
More informationMATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 2: Countability and Cantor Sets
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 2: Countability and Cantor Sets Countable and Uncountable Sets The concept of countability will be important in this course
More informationPart II Logic and Set Theory
Part II Logic and Set Theory Theorems Based on lectures by I. B. Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationMathematics 220 Workshop Cardinality. Some harder problems on cardinality.
Some harder problems on cardinality. These are two series of problems with specific goals: the first goal is to prove that the cardinality of the set of irrational numbers is continuum, and the second
More informationProf. Ila Varma HW 8 Solutions MATH 109. A B, h(i) := g(i n) if i > n. h : Z + f((i + 1)/2) if i is odd, g(i/2) if i is even.
1. Show that if A and B are countable, then A B is also countable. Hence, prove by contradiction, that if X is uncountable and a subset A is countable, then X A is uncountable. Solution: Suppose A and
More informationA Logician s Toolbox
A Logician s Toolbox 461: An Introduction to Mathematical Logic Spring 2009 We recast/introduce notions which arise everywhere in mathematics. All proofs are left as exercises. 0 Notations from set theory
More informationSets, Models and Proofs. I. Moerdijk and J. van Oosten Department of Mathematics Utrecht University
Sets, Models and Proofs I. Moerdijk and J. van Oosten Department of Mathematics Utrecht University 2000; revised, 2006 Contents 1 Sets 1 1.1 Cardinal Numbers........................ 2 1.1.1 The Continuum
More informationLebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures.
Measures In General Lebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures. Definition: σ-algebra Let X be a set. A
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.
MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection
More informationFinal Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is
1. Describe the elements of the set (Z Q) R N. Is this set countable or uncountable? Solution: The set is equal to {(x, y) x Z, y N} = Z N. Since the Cartesian product of two denumerable sets is denumerable,
More information06 Recursive Definition and Inductive Proof
CAS 701 Fall 2002 06 Recursive Definition and Inductive Proof Instructor: W. M. Farmer Revised: 30 November 2002 1 What is Recursion? Recursion is a method of defining a structure or operation in terms
More informationTutorial on Axiomatic Set Theory. Javier R. Movellan
Tutorial on Axiomatic Set Theory Javier R. Movellan Intuitively we think of sets as collections of elements. The crucial part of this intuitive concept is that we are willing to treat sets as entities
More informationCMSC 27130: Honors Discrete Mathematics
CMSC 27130: Honors Discrete Mathematics Lectures by Alexander Razborov Notes by Geelon So, Isaac Friend, Warren Mo University of Chicago, Fall 2016 Lecture 1 (Monday, September 26) 1 Mathematical Induction.................................
More informationMATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X.
MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X. Notation 2 A set can be described using set-builder notation. That is, a set can be described
More informationSelected problems from past exams
Discrete Structures CS2800 Prelim 1 s Selected problems from past exams 1. True/false. For each of the following statements, indicate whether the statement is true or false. Give a one or two sentence
More informationDefinition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.
4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be
More informationInfinite constructions in set theory
VI : Infinite constructions in set theory In elementary accounts of set theory, examples of finite collections of objects receive a great deal of attention for several reasons. For example, they provide
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationClearly C B, for every C Q. Suppose that we may find v 1, v 2,..., v n
10. Algebraic closure Definition 10.1. Let K be a field. The algebraic closure of K, denoted K, is an algebraic field extension L/K such that every polynomial in K[x] splits in L. We say that K is algebraically
More informationMATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals
MATH 02 INTRODUCTION TO MATHEMATICAL ANALYSIS Properties of Real Numbers Some Fundamentals The whole course will be based entirely on the study of sequence of numbers and functions defined on the real
More informationWeek Some Warm-up Questions
1 Some Warm-up Questions Week 1-2 Abstraction: The process going from specific cases to general problem. Proof: A sequence of arguments to show certain conclusion to be true. If... then... : The part after
More informationMath 109 September 1, 2016
Math 109 September 1, 2016 Question 1 Given that the proposition P Q is true. Which of the following must also be true? A. (not P ) or Q. B. (not Q) implies (not P ). C. Q implies P. D. A and B E. A, B,
More informationMath 280A Fall Axioms of Set Theory
Math 280A Fall 2009 1. Axioms of Set Theory Let V be the collection of all sets and be a membership relation. We consider (V, ) as a mathematical structure. Analogy: A group is a mathematical structure
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationSouslin s Hypothesis
Michiel Jespers Souslin s Hypothesis Bachelor thesis, August 15, 2013 Supervisor: dr. K.P. Hart Mathematical Institute, Leiden University Contents 1 Introduction 1 2 Trees 2 3 The -Principle 7 4 Martin
More informationSlow P -point Ultrafilters
Slow P -point Ultrafilters Renling Jin College of Charleston jinr@cofc.edu Abstract We answer a question of Blass, Di Nasso, and Forti [2, 7] by proving, assuming Continuum Hypothesis or Martin s Axiom,
More information