Lecture 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, 2007; Time: 15:24 1 Textbook: R. J. Bond and W. J. Keane, An Introduction to Abstract Mathematics, Brooks/Cole, 1999

Contents 1 Logic 1 1.1 LECTURE 1. Statements...................... 1 1.1.1 Proof............................ 1 1.1.2 Statements......................... 2 1.1.3 Quantifiers......................... 2 1.1.4 Negations.......................... 3 1.1.5 Homework......................... 4 1.2 LECTURE 2. Compound Statements................ 5 1.2.1 Conjunctions and Disjunctions............... 5 1.2.2 Truth Tables........................ 5 1.2.3 Negating Conjunctions and Disjunctions.......... 6 1.2.4 Logically Equivalent Statements.............. 7 1.2.5 Tautologies and Contradictions............... 8 1.2.6 Homework......................... 9 1.3 LECTURE 3. Implications..................... 10 1.3.1 Truth Table for an Implication............... 11 1.3.2 Proving Statements Containing Implications........ 11 1.3.3 Negating an Implication: Counterexamples........ 12 1.3.4 Necessary and Sufficient Conditions............ 13 1.3.5 Homework......................... 14 1.4 LECTURE 4. Contrapositive and Converse............ 15 1.4.1 Contrapositive....................... 15 1.4.2 Converse.......................... 16 1.4.3 Biconditional........................ 16 1.4.4 Proof by Contradiction................... 17 1.4.5 Homework......................... 18 I

II CONTENTS 2 Sets 19 2.1 LECTURE 5. Sets and Subsets................... 19 2.1.1 The Notion of a Set..................... 19 2.1.2 Subsets........................... 21 2.1.3 Complements........................ 22 2.1.4 Homework......................... 22 2.2 LECTURE 6. Combining Sets................... 23 2.2.1 Unions and Intersections.................. 23 2.2.2 De Morgan s Laws..................... 24 2.2.3 Cartesian Products..................... 24 2.2.4 Homework......................... 24 2.3 LECTURE 7. Collection of Sets.................. 25 2.3.1 Power Set.......................... 25 2.3.2 Indexing Sets........................ 25 2.3.3 Partitions.......................... 27 2.3.4 The Pigeonhole Principle.................. 27 2.3.5 Cantor Set......................... 28 2.3.6 Homework......................... 29 3 Functions 31 3.1 LECTURE 8. Definition and Basic Properties........... 31 3.1.1 Image of a Function.................... 32 3.1.2 Inverse Image........................ 34 3.1.3 Homework......................... 35 3.2 LECTURE 9. Surjective and Injective Functions.......... 36 3.2.1 Surjective Functions.................... 36 3.2.2 Injective Functions..................... 36 3.2.3 Bijective Functions..................... 37 3.2.4 Homework......................... 37 3.3 LECTURE 10. Composition and Inverse Functions........ 38 3.3.1 Composition of Functions................. 38 3.3.2 Inverse Functions...................... 39 3.3.3 Homework......................... 40 4 Binary Operations and Relations 41 4.1 LECTURE 11. Binary Operations................. 41 4.1.1 Associate and Commutative Laws............. 41 4.1.2 Identities.......................... 42 absmath.tex; June 11, 2007; 15:24; p. 1

CONTENTS III 4.1.3 Inverses........................... 43 4.1.4 Closure........................... 43 4.1.5 Groups........................... 44 4.1.6 Homework......................... 44 4.2 LECTURE 12. Equivalence Relations............... 45 4.2.1 Relations.......................... 45 4.2.2 Properties of Relations................... 45 4.2.3 Equivalence Relations................... 46 4.2.4 Equivalence Classes.................... 46 4.2.5 Partial and Linear Ordering................ 48 4.2.6 Homework......................... 49 5 The Integers 51 5.1 LECTURE 13. Axioms and Basic Properties............ 51 5.1.1 The Axioms of the Integers................ 51 5.1.2 Inequalities......................... 53 5.1.3 The Well-Ordering Principle................ 54 5.1.4 Homework......................... 55 5.2 LECTURE 14. Induction...................... 56 5.2.1 Induction: A Method of Proof............... 56 5.2.2 Other Forms of Induction................. 56 5.2.3 The Binomial Theorem.................. 58 5.2.4 Homework......................... 59 5.3 LECTURE 15. The Division Algorithm.............. 60 5.3.1 Divisors and Greatest Common Divisors......... 60 5.3.2 Euclidean Algorithm.................... 62 5.3.3 Relatively Prime Integers................. 63 5.3.4 Homework......................... 64 5.4 LECTURE 16. Primes and Unique Factorization......... 65 5.4.1 Prime Numbers...................... 65 5.4.2 Unique Factorization.................... 66 5.4.3 Euclid Theorem....................... 68 5.4.4 Homework......................... 68 5.5 LECTURE 17. Congruences.................... 69 5.5.1 Congruences and Their Properties............. 69 5.5.2 The Set of Congruence Classes.............. 71 5.5.3 Homework......................... 75 absmath.tex; June 11, 2007; 15:24; p. 2

IV CONTENTS 6 Infinite Sets 77 6.1 LECTURE 19. Countable Sets................... 77 6.1.1 Numerically Equivalent Sets................ 77 6.1.2 Countable Sets....................... 78 6.1.3 Unions of Countable Sets................. 82 6.1.4 The Rationals are Countable................ 83 6.1.5 Cartesian Products of Countable Sets........... 83 6.1.6 Homework......................... 84 6.2 LECTURE 20. Uncountable Sets.................. 85 6.2.1 Uncountable Sets...................... 85 6.2.2 Cantor Theorem...................... 86 6.2.3 Continuum Hypothesis................... 87 6.2.4 Schroeder-Bernstein Theorem............... 88 6.2.5 Homework......................... 92 6.3 LECTURE 21. Collections of Sets................. 93 6.3.1 Russell s Paradox..................... 93 6.3.2 Countable Unions of Countable Sets........... 93 6.3.3 Homework......................... 95 7 The Real and Complex Numbers 97 7.1 LECTURE 22. Fields........................ 97 7.1.1 Fields............................ 97 7.1.2 Elementary Properties of Fields.............. 98 7.1.3 Ordered Fields....................... 99 7.1.4 Finite Fields........................ 101 7.1.5 Homework......................... 103 7.2 LECTURE 23. The Real Numbers................. 104 7.2.1 Bounded Sets........................ 104 7.2.2 Least Upper Bound and Greatest Lower Bound...... 104 7.2.3 The Archimedean Principle................ 105 7.2.4 Incompleteness of Rationals................ 106 7.2.5 Homework......................... 107 7.3 LECTURE 24. The Complex Numbers............... 108 7.3.1 Complex Numbers..................... 108 7.3.2 Conjugation and Absolute Value............. 109 7.3.3 Solutions of Equations................... 110 7.3.4 Polar Form......................... 110 7.3.5 Complex Roots....................... 111 absmath.tex; June 11, 2007; 15:24; p. 3

CONTENTS V 7.3.6 Homework......................... 112 Bibliography 113 absmath.tex; June 11, 2007; 15:24; p. 4

VI CONTENTS absmath.tex; June 11, 2007; 15:24; p. 5

Chapter 1 Logic 1.1 LECTURE 1. Statements 1.1.1 Proof Mathematics is an attempt to determine which statements are true and which are not. Inductive reasoning Deductive reasoning Conjecture is a not verified statement Proof is the verification of a statement. Axioms are statements that are accepted as given. Propositions are the logical deductions from the axioms. Theorems are particularly important propositions. Proof is the logic, the arguments, used to make deductions Example An integer m is a multiple of an integer n if m = kn for some integer k. 1

2 CHAPTER 1. LOGIC 1.1.2 Statements Definition 1.1.1 true or false. A statement is any declarative sentence that is either Example A variable is a symbol that stands for an undetermined number. Definition 1.1.2 An open sentence is any declarative sentence containing one or more variables that is not a statement but becomes a statement when the variables are assigned values. Notation: P(x), P(x, y) 1.1.3 Quantifiers A universal quantifier is a phrase for every, for all, etc Notation: Example An existential quantifier is a phrase there is, there exists, etc Notation: Example A bound variable is a variable to which a quantifier is applied. A variable that is not bound is a free variable. Notation: The symbol is read such that Remark: The order in which quantifiers appear in a statement is important. Example absmath.tex; June 11, 2007; 15:24; p. 6

1.1. LECTURE 1. STATEMENTS 3 1.1.4 Negations Example Definition 1.1.3 false. The negation of a statement P is the statement P is Notation: The negation of P is denoted by the symbol P, read not P Alternative ways to express P: P is not true, or It is not true that P Remark. Exactly one of P and P is true; the other is false. Example Negation of a statement with a universal quantifier Example Negation of a statement with an existential quantifier Example Basic Rules of negating statements with quantifiers absmath.tex; June 11, 2007; 15:24; p. 7

4 CHAPTER 1. LOGIC The negation of the statement For all x, P(x) is the statement For some x, P(x) ( x, P(x)) x P(x) The negation of the statement For some x, P(x) is the statement For all x, P(x) ( x, P(x)) x, P(x) Example ( x, y P(x, y)) x y, P(x, y) Definition 1.1.4 (Archimedean Principle) For every real number x, there is an integer n such that n > x. 1.1.5 Homework Read Introduction for the Student and to the Chapter 1; pp. xv-xix, 1. Reading: Sect 1.1 Exercises: 1.1[9,11,D5] absmath.tex; June 11, 2007; 15:24; p. 8

1.2. LECTURE 2. COMPOUND STATEMENTS 5 1.2 LECTURE 2. Compound Statements 1.2.1 Conjunctions and Disjunctions Compound statements are statements built up from two or more statements. Definition 1.2.1 is the statement Notation: The conjunction of a statement P and a statement Q Both P and Q are true. P Q, read P and Q Definition 1.2.2 is the statement The disjunction of a statement P and a statement Q P is true or Q is true. Notation: P Q, read P or Q Remarks. Similarly for open sentences, P(x) Q(x) etc. Example 1.2.2 Truth Tables Statement forms are expressions of the form P Q, P Q, or P, where P and Q are variables representing unspecified statements. Statement forms are not statements but they become statements when the variables P and Q are specified, that is, replaced by statements. Truth tables are tables of truth values of the statement forms. absmath.tex; June 11, 2007; 15:24; p. 9

6 CHAPTER 1. LOGIC Example P Q P Q T T T T F F F T F F F F Remarks Example P Q P Q T T T T F T F T T F F F P P T F F T 1.2.3 Negating Conjunctions and Disjunctions Example P Q P Q P Q P Q T T F F T T T F F T F T F T T F F T F F T T F F P Q (P Q) (P Q) P Q P Q T T F F F F T F T F T F F T T F T F F F T T T T That is, (P Q) P Q (P Q) P Q These statement forms mean the same thing. Example absmath.tex; June 11, 2007; 15:24; p. 10

1.2. LECTURE 2. COMPOUND STATEMENTS 7 1.2.4 Logically Equivalent Statements Statements and statement forms are logically equivalent if they are expressed in different ways but mean the same thing. Definition 1.2.3 Two statements are logically equivalent or just equivalent if they are both true or both false. Definition 1.2.4 Two statement forms are logically equivalent if the substitution of statements for the variables in the forms always yields logically equivalent statements. Notation. P Q read P is equivalent to Q Remark. If two statement forms have the same truth tables then they are logically equivalent. Examples. 1. (P Q) P Q 2. (P Q) P Q 3. ( x, P(x) x ( P(x)) 4. ( x P(x) x, ( P(x)) 5. ( x, (P(x) Q(x)) x (( P(x)) ( Q(x))) 6. ( x, (P(x) Q(x)) x (( P(x)) ( Q(x))) 7. ( x (P(x) Q(x)) x, (( P(x)) ( Q(x))) 8. ( x (P(x) Q(x)) x, (( P(x)) ( Q(x))) Examples. The statements and are not equivalent, that is For all x, P(x) or Q(x) For all x, P(x) or for all x, Q(x) x, (P(x) Q(x)) x, P(x) x, Q(x) absmath.tex; June 11, 2007; 15:24; p. 11

8 CHAPTER 1. LOGIC Proposition. Proof: Truth table. Example x (P(x) Q(x)) ( x P(x)) ( x Q(x))). One can construct compound statements and statement forms from three or more statements Proposition. Proof: (P Q) (P R) P (Q R) P Q R P Q P R (P Q) (P R) Q R P (Q R) T T T T T T T T T T F F T F T T T F T F T T T T T F F F F F F F F T T F F F T F F T F F F F T F F F T F F F T F F F F F F F F F 1.2.5 Tautologies and Contradictions Tautology is a statement form that is always true no matter what statements are substituted for the variables. Remark. Each of the truth table values of a tautology is true. Example. P P is always true. Contradiction is a statement form that is always false no matter what statements are substituted for the variables. Remark. Each of the truth table values of a contradiction is false. Example. P P is always false. Negation of a tautology is a contradiction and vice versa. absmath.tex; June 11, 2007; 15:24; p. 12

1.2. LECTURE 2. COMPOUND STATEMENTS 9 1.2.6 Homework Reading: Sect 1.2 Exercises: 1.2[4,11,12] absmath.tex; June 11, 2007; 15:24; p. 13

10 CHAPTER 1. LOGIC 1.3 LECTURE 3. Implications Theorems follow logically from prior propositions Propositions follow from axioms Axioms are statements that are taken without proof and that serve as the starting point Implications are statements of the form If..., then..., or For all..., if..., then.... Example The premise or the assumption of the statement is the if part. The conclusion is the then part. Definition 1.3.1 is the statement Let P and Q be statements. The implication P Q If P is true, then Q is true. Notation. P Q (read P implies Q ) Example absmath.tex; June 11, 2007; 15:24; p. 14

1.3. LECTURE 3. IMPLICATIONS 11 1.3.1 Truth Table for an Implication In the implication P Q, P might be unrelated to Q at all. P does not have to have caused Q The implication P Q means that in all circumstances under which P is true, Q is also true. In other words, implication P Q means that whenever P happens, Q also happens. The implication P Q is false only in the case if P is true and Q is false. The implication P Q cannot be false if P is false, even if Q is false. A false statement implies anything! Truth values of the statement form P Q: P Q P Q T T T T F F F T T F F T Examples. 1.3.2 Proving Statements Containing Implications Let P(x) and Q(x) be open sentences. The expression P(x) Q(x) is an open sentence that becomes a statement when the variable x is assigned a value a. Note that the expression x, P(x) Q(x) is a statement. We want to prove the statement x, P(x) Q(x). Proof: Assume that if the variable x is assigned the value a, then P(a) is true and proceed to prove Q(a). P(a) is the hypothesis absmath.tex; June 11, 2007; 15:24; p. 15

12 CHAPTER 1. LOGIC Q(a) is the conclusion Example Remark. Let P(n) be an open sentence. Once we say, Let n be an integer, then P(n) is a statement. Example 1.3.3 Negating an Implication: Counterexamples In order to prove that a statement containing an implication is false it is necessary to prove that its negation is true. Example. The only way for an implication x, P(x) Q(x) to be false is that there is an assigned value a of the variable x such that P(a) is true and Q(a) is false. A value of a variable served to disprove a statement with a universal quantifier is a counterexample. Proposition. If P and Q are statement forms, then Proof: Find truth tables. (P Q) P Q If P(x) and Q(x) are open sentences, the negation of the statement is the statement or For all x, P(x) Q(x) There exists x such that P(x) Q(x), For some x, P(x) is true and Q(x) is false. absmath.tex; June 11, 2007; 15:24; p. 16

1.3. LECTURE 3. IMPLICATIONS 13 Formally ( x, P(x) Q(x)) x (P(x) Q(x)) The value assigned to the variable x that makes P(x) true and Q(x) false is a counterexample to the statement For all x, P(x) Q(x). Negation of an implication is not an implication! 1.3.4 Necessary and Sufficient Conditions In the implication P Q (if P is true, then Q is true) P is a sufficient condition for Q. In order for Q to be true it is sufficient that P be true. If P Q is true, (it is true that if P is true then Q is true), then Q is a necessary condition for P. Q must be true in order for P to be true. In other words, if Q is false, then P is false. Proposition. The statement is equivalent to the statement Q P P Q. ( Q P) (P Q) Proof: Truth table. Remarks. If P Q is true then it is not necessary that P be true in order for Q to be true. absmath.tex; June 11, 2007; 15:24; p. 17

14 CHAPTER 1. LOGIC Also, Q is not a sufficient condition for P. That is, even if Q is true, P may be false. Example 1.3.5 Homework Reading: Sect 1.3 Exercises: 1.3[3,13,15] absmath.tex; June 11, 2007; 15:24; p. 18

1.4. LECTURE 4. CONTRAPOSITIVE AND CONVERSE 15 1.4 LECTURE 4. Contrapositive and Converse 1.4.1 Contrapositive The statements P Q and Q P are logically equivalent (P Q) ( Q P) Check the truth tables To prove the statement P Q is the same as to verify the statement Q P This means: whenever Q is false, then P is false. In other words: if P is true, then Q cannot be false and, therefore, Q is true. Example Definition 1.4.1 Let P and Q be two statements. The statement Q P is the contrapositive of the statement P Q. Example absmath.tex; June 11, 2007; 15:24; p. 19

16 CHAPTER 1. LOGIC 1.4.2 Converse Example Definition 1.4.2 The statement Q P is the converse of the statement P Q. Example The statements P Q and its converse Q P are not necessarily logically equivalent! (P Q) (Q P) If the implication P Q is true, its converse Q P could be false. Example 1.4.3 Biconditional If the statement P Q and its converse Q P are both true, then the statements P and Q are either both true or both false. Then P is a necessary and sufficient condition for Q. Thus, P and Q are logically equivalent. Definition 1.4.3 The statement P Q is the statement (P Q) (Q P). The symbol is called the biconditional. Notation. The statement P Q is read P if and only if Q. absmath.tex; June 11, 2007; 15:24; p. 20

1.4. LECTURE 4. CONTRAPOSITIVE AND CONVERSE 17 The word iff is the shorthand for the phrase if and only if. Theorem 1.4.1 Let n be an integer. Then n is even if and only if n 2 is even. Theorem 1.4.2 Let n be an integer. The following are equivalent statements: 1. n is even. Proof: 2. n 2 is even. Proposition. The statements and are logically equivalent. (P Q) ( P Q) (P Q) ( P Q) Proof: By examining the truth tables. Corollary 1.4.1 Let n be an integer. Then n is odd if and only if n 2 is odd. Proof: Example 1.4.4 Proof by Contradiction In order to prove an implication P Q one can prove its contrapositive Q P. Proof by contradiction. absmath.tex; June 11, 2007; 15:24; p. 21

18 CHAPTER 1. LOGIC Proving means that we Q P assume that Q is false, and prove that P is false, getting a contradiction that P and P cannot both be true. To prove a statement P by contradiction means to assume that P is false, and deduce a false statement Q from it. Theorem 1.4.3 Let P be a statement and Q be a false statement. Then the statement is logically equivalent to P. Proof: Example P Q 1.4.5 Homework Reading: Sect 1.4 Exercises: 1.4[2,17,21] absmath.tex; June 11, 2007; 15:24; p. 22

Chapter 2 Sets 2.1 LECTURE 5. Sets and Subsets 2.1.1 The Notion of a Set The set is a basic undetermined concept which cannot be defined formally. The set is a collection of objects called the elements. Notation. Example Notation. x A (read x is in A). N = Z + the set of positive integers (natural numbers), Z the set of integers, Q = { m m, n Z, n 0} the set of rational numbers numbers, n R + the set of positive real numbers, R the set of real numbers, C the set of complex numbers. Example 19

20 CHAPTER 2. SETS Description of the set of all elements of the set U such that the open sentence P(x) is true statement A = {x U P(x)} So, A is the truth set of the open sentence P(x). The set U is a fixed set called a universal set. Example. The set of multiples of n nz = {m Z m = nk for some k Z} = {nk k Z} Definition 2.1.1 A set I of real numbers is called an interval if I, I contains more than one element, and for every x, y I such that x < y, if z a real number such that x < z < y, then z I. Closed bounded interval Open bounded interval [a, b] = {x R a x b} (a, b) = {x R a < x < b} Half-open, half-closed bounded intervals (a, b] = {x R a < x b}, [a, b) = {x R a x < b} Unbounded intervals (, b] = {x R x b}, (a, ) = {x R a < x}, (, b) = {x R a x < b}, [a, ) = {x R a x}, Example Sets containing finitely many elements are called finite sets. The number of elements of a finite set is called the cardinality of the set. Notation. A Sets containing infinitely many elements are called infinite sets. Example absmath.tex; June 11, 2007; 15:24; p. 23

2.1. LECTURE 5. SETS AND SUBSETS 21 2.1.2 Subsets Example Definition 2.1.2 Let A and B be sets. A is a subset of B if every element of A is also an element of B. If the set B has an element that is not an element of its subset A then the subset A is called a proper subset. Notation. Example A B, (A is a subset of B). A B (A is a proper subset of B) Let U be an universal set. Then A B is the statement Example x U, if x A, then x B. Proposition 2.1.1 Let A, B and C be sets. If A B and B C, then A C. Proof: Example Definition 2.1.3 elements. Notation. A = B. Two sets A and B are equal if they have the same Two sets A and B are equal if and only if every element of A is an element of B and every element of B is an element of A (A = B) (A B A) The statement A B is false if there is an element of A such that it is not an element of B. Example absmath.tex; June 11, 2007; 15:24; p. 24

22 CHAPTER 2. SETS 2.1.3 Complements Definition 2.1.4 the set Let A and B be sets. The complement of A in B is B A = {x B x A}. Example The set A does not need to be a subset of the set B. Let U be a universal set. The the complement of A in U is simply called the complement of A. Notation. Ā = U A. The set that contains no elements is called the empty set. Notation. If U is an universal set, then = U and Ū =. Theorem 2.1.1 Let U be a universal set and A and B be sets contained in U. Then Proof: Example 2.1.4 Homework Reading: Sect 2.1 Exercises: 2.1[1,10,14] A B iff B Ā. absmath.tex; June 11, 2007; 15:24; p. 25

2.2. LECTURE 6. COMBINING SETS 23 2.2 LECTURE 6. Combining Sets 2.2.1 Unions and Intersections Definition 2.2.1 The union of a set A and a set B is the set A B = {x x A x B}. Definition 2.2.2 The intersection of a set A and a set B is the set A B = {x x A x B}. Example Two sets are disjoint if they do not have common elemets, that is their intersection is the empty set, Venn diagrams A B =. Theorem 2.2.1 Let A, B and C are sets. Then 1. A B = B A, 2. A B = B A, 3. (A B) C = A (B C), 4. (A B) C = A (B C), 5. A (A B), 6. (A B) A, 7. A, 8. A = A, 9. A =. Proof: Proposition. Let A, B U. Then absmath.tex; June 11, 2007; 15:24; p. 26

24 CHAPTER 2. SETS 1. A B = A B. 2. A B A B = B Proof: Theorem 2.2.2 Let A, B and C be sets. Then 1. A (B C) = (A B) (A C), Proof: 2. A (B C) = (A B) (A C). 2.2.2 De Morgan s Laws Example Theorem 2.2.3 Let U be a universal set and A, B U. Then Proof: 1. A B = Ā B, 2. A B = Ā B. 2.2.3 Cartesian Products Definition 2.2.3 set The Cartesian product of a set A and a set B is the A B = {(a, b) a A, b B}. The Cartesian product is the set of ordered pairs (a, b), where the first element is from the set A and the second element is from B. Example 2.2.4 Homework Reading: Sect 2.2 Exercises: 2.2[22,24] absmath.tex; June 11, 2007; 15:24; p. 27

2.3. LECTURE 7. COLLECTION OF SETS 25 2.3 LECTURE 7. Collection of Sets 2.3.1 Power Set Example Definition 2.3.1 The power set of a set A is the set of all subsets of A. Notation. P(A) = {X X A}. Note that P(A) and A P(A). If the set A is finite, then the cardinality of the power set is P(A) = 2 A Example 2.3.2 Indexing Sets Example The indexing set is the set of subscripts that are used to distinguish the sets in a collection of sets. Notation. where I = {1, 2,..., n}. S = {A i i I} = {A i } i I, Union of sets in a collection of sets n A i = {x U i I x A i } i=1 absmath.tex; June 11, 2007; 15:24; p. 28

26 CHAPTER 2. SETS Intersection of sets in a collection of sets n A i = {x U x A i i I} i=1 Example Infinite collection of sets. Union of sets in a collection of sets A i = {x U i N x A i } i=1 Intersection of sets in a collection of sets A i = {x U x A i i N} i=1 Example The collection of sets S = {A i } i I is increasing (or an ascending chain) if A 1 A 2 A 3 The collection of sets S = {A i } i I is decreasing (or an descending chain) if A 3 A 2 A 1. Example Let I be an indexing set and S = {A i } i I be a collec- Definition 2.3.2 tion of sets. 1. The union of the collection is A i = {a a A for some i I} i I 2. The intersection of the collection is A i = {a a A for all i I} i I absmath.tex; June 11, 2007; 15:24; p. 29

2.3. LECTURE 7. COLLECTION OF SETS 27 Notation. Let S be a collection of sets. Then the union and the intersection are A S A and A S A. 2.3.3 Partitions Definition 2.3.3 P(A) such that A partition of a set A is a subset P of the power set 1. if X P, thenx, 2. X P X = A, 3. if X, Y P and X Y, then X Y =. That is, a partition is a collection of non-empty disjoint subsets of A that cover the set A. Remark. A partition of a set divides the set into different disjoint nonempty subsets so that every element of the set is in one of the subsets and no element is in more than one. Example 2.3.4 The Pigeonhole Principle Theorem 2.3.1 Let A and B be finite disjoint sets. Then A B = A + B Proof: Rigorous proof later. Corollary 2.3.1 Let S = {A i } n i=1 be a collection of finite mutually disjoint sets. Then n n A i = A i i=1 i=1 absmath.tex; June 11, 2007; 15:24; p. 30

28 CHAPTER 2. SETS Corollary 2.3.2 Let A and B be finite sets. Then A B = A + B A B Pigeonhole Principle. If n objects are placed in k containers and n > k, then at least one container will have more than one object in it. Let S = {A i } n i=1 n i=1 A i. be a collection of finite mutually disjoint sets and A = If A = k and k > n, then, for some i, A i 2. Example Theorem 2.3.2 Let S = {A i } n i=1 be a collection of finite mutually disjoint sets and A = n i=1 A i. If A > nr for some positive integer r, then, for some i, A i r + 1. Proof: Exercise. Example 2.3.5 Cantor Set Let A = [0, 1]. Let A 1 = [0, 1 3 ] [ 2 3, 1] Let A 2 = [0, 1 9 ] [ 2 9, 3 9 ] [ 6 9, 7 9 ] [ 8 9, 1] Define A n by removing the middle third from each of the closed intervals that make the set A n 1. Each A n is the union of 2 n closed intervals Each interval that makes A n has length 1 3 n The total length of all intervals that make A n is ( 2 3 The collection of sets {A n } n=1 forms a descending chain ) n A n+1 A n, n N absmath.tex; June 11, 2007; 15:24; p. 31

2.3. LECTURE 7. COLLECTION OF SETS 29 The Cantor set is defined by C = n=1 A n The endpoints of all intervals in each of the sets A n are in C 0, 1, 1 3, 2 3, 1 9, 2 9, 7 9, 8 9, C Cantor set is infinite. Cantor set is bounded. The size (length) of the Cantor set is zero. Since the length of C is smaller than the length of A n for any n N, which is ( 2 3 )n. Thus it is smaller than ( 2 3 )n for any n, and is, therefore, zero. Cantor set is uncountable. So, it is larger than the set of integers. 2.3.6 Homework Reading: Sect 2.3 Exercises: 2.3[16,26,28] absmath.tex; June 11, 2007; 15:24; p. 32

30 CHAPTER 2. SETS absmath.tex; June 11, 2007; 15:24; p. 33

Chapter 3 Functions 3.1 LECTURE 8. Definition and Basic Properties Definition 3.1.1 Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element of the set A one and only one element of the set B. The set A is the domain of f and the set B the codomain. Notation. f : A B and for each a A, f (a) = b or a f b. Example The identity function on A is the function i A : A A defined by i A (x) = x for any x A. It is also denoted by Id A or id A. 31

32 CHAPTER 3. FUNCTIONS 3.1.1 Image of a Function Definition 3.1.2 Let f : A B be a function and X A. The image of the set X under the function f is the set Diagram. Remark. f (X) = {y B y = f (x) for some x X} f ( ) = for any function f. Definition 3.1.3 Let f : A B be a function. The image (or the range) of the function f is the set Remark. Diagram. Im( f ) = f (A) = {y B y = f (x) for some x A} Im( f ) B for any function f. Definition 3.1.4 Let f : A B be a function. The graph of the function f is the set Diagram. Remark. Γ( f ) = {(a, b) A B b = f (a), a A} Γ( f ) A B. Remark. The functions f : A B and g : A f (A) such that g(x) = f (x) for any x A are different if f (A) B. Definition 3.1.5 Two functions f and g are equal if they have the same domain and the same codomain and if f (x) = g(x) for any x in the domain. absmath.tex; June 11, 2007; 15:24; p. 34

3.1. LECTURE 8. DEFINITION AND BASIC PROPERTIES 33 Examples Sine function sin : R R, x sin sin(x) Greatest integer function. f : R Z, f (x) = [x] is the greatest integer x Finding the image of a function. If f : A B is a function and y B, then y Im( f ) if and only if there exists an element x A such that f (x) = y. Example. Theorem 3.1.1 Intermediate Value Theorem. Let A, B R be subsets of real numbers and f : A B be a function. Let [a, b] A and f be continuous on the interval [a, b]. If y is a number between f (a) and f (b), then there is a real number x [a, b] such that f (x) = y. Without proof. Graphical illustration. Example Proposition 3.1.1 Let A and B be sets and X and Y be subsets of A such that Let f : A B be a function. Then Proof: Easy. X Y A. f (X) f (Y). The converse of this proposition is false. Example. absmath.tex; June 11, 2007; 15:24; p. 35

34 CHAPTER 3. FUNCTIONS Proposition 3.1.2 Let A and B be sets and X and Y be subsets of A. Let f : A B be a function. Then Proof: Example. 1. f (X Y) = f (X) f (Y). 2. f (X Y) f (X) f (Y). 3.1.2 Inverse Image Definition 3.1.6 Let A and B be sets and W be a subset of B. Let f : A B be a function. The the inverse image of the set W with respect to f is the set Diagram. f 1 (W) = {x A f (x) W} The inverse image f 1 (W) is the set of all elements of the domain of f that are mapped to elements of W. Remarks. Inverse image is a subset of the domain. f 1 (W) A W is not necessarily a subset of the image Im( f ) of the function f. The symbol f 1 does not refer to the inverse function, which might not even exist. Example Proposition 3.1.3 Let A and B be sets and W and Z be subsets of B. Let f : A B be a function. Then Proof: 1. f 1 (W Z) = f 1 (W) f 1 (Z) 2. f 1 (W Z) = f 1 (W) f 1 (Z) absmath.tex; June 11, 2007; 15:24; p. 36

3.1. LECTURE 8. DEFINITION AND BASIC PROPERTIES 35 Proposition 3.1.4 Let A and B be sets and f : A B be a function. If A is a finite set, then Im( f ) is a finite set and Proof: Easy. Example 3.1.3 Homework Reading: Sect 3.1 Exercises: 3.1[2,4,11] Im( f ) A. absmath.tex; June 11, 2007; 15:24; p. 37

36 CHAPTER 3. FUNCTIONS 3.2 LECTURE 9. Surjective and Injective Functions. 3.2.1 Surjective Functions Definition 3.2.1 Let f : A B be a function. Then f is surjective (or a surjection) if the image of f is equal to the codomain of f, that is Im( f ) = B. To prove that a function is surjective one needs to prove that for any element y in the codomain B there is an element in the domain x A such that f (x) = y. Diagram. Surjection is also called an onto mapping. Example Let A and B be sets. The function π : A B A defined by is a projection of A B onto A. π(a, b) = a To prove that the function f : A B is not surjective one needs to show that there exists an element y B in the codomain B such that for any element x A in the domain, f (x) y. Examples. 3.2.2 Injective Functions Example Definition 3.2.2 Let f : A B be a function. Then f is injective (or an injection) if for any a 1, a 2 A, if a 1 a 2, then f (a 1 ) f (a 2 ). absmath.tex; June 11, 2007; 15:24; p. 38

3.2. LECTURE 9. SURJECTIVE AND INJECTIVE FUNCTIONS. 37 An injection takes different elements of the domain A to different elements of the codomain B. Injection is also called an one-to-one mapping. Diagram. Example To prove that a function f : A B is injective one needs to prove that Examples. for all a 1, a 2 A, if f (a 1 ) = f (a 2 ), then a 1 = a 2. To prove that a function f : A B is not injective one needs to show that there exist a 1, a 2 A such that a 1 a 2 and f (a 1 ) = f (a 2 ). (In other words, f maps two different elements of A to the same element of B.) Example. 3.2.3 Bijective Functions Definition 3.2.3 A function that is both injective and surjective is bijective (or a bijection). Examples. Let A be a set. A bijection f : A A is a permuta- Definition 3.2.4 tion of A. Example 3.2.4 Homework Reading: Sect 3.2 Exercises: 3.2[2,13,17] absmath.tex; June 11, 2007; 15:24; p. 39

38 CHAPTER 3. FUNCTIONS 3.3 LECTURE 10. Composition and Inverse Functions Definition 3.3.1 Let A and B be nonempty sets. The sets of all functions from A to B is denoted by If A = B, then we denote F(A, B) = { f f : A B}. F(A, A) = F(A). 3.3.1 Composition of Functions Definition 3.3.2 Let A, B, and C be nonempty sets, and let f F(A, B) and g F(B, C). The composition of f and g is the function g f F(A, C) defined by (g f )(x) = g( f (x)), x A Notation. The composition g f is also denoted simply by g f. If A = B = C, then both g f and f g are defined. Examples. Proposition 3.3.1 Let f : A B be a function. Then f i A = i B f = f. Proof: Exercise. Proposition 3.3.2 Let f F(A, B) and g F(B, C). Then: 1. If f and g are surjections, then g f is a surjection. 2. If f and g are injections, then g f is an injection. Proof: 3. If f and g are bijections, then g f is a bijection. absmath.tex; June 11, 2007; 15:24; p. 40

3.3. LECTURE 10. COMPOSITION AND INVERSE FUNCTIONS 39 Corollary 3.3.1 Let A be a nonempty set. Let f and g be permutations of A. Then g f is a permutation of A. Proof: Exercise. Proposition 3.3.3 Let A, B, C and D be nonempty sets. Let f F(A, B), g F(B, C), h F(C, D). Then Proof: (h g) f = h (g f ). Notation. One can write without ambiguity simply h g f = (h g) f = h (g f ). If we have n functions f 1, f 2,..., f n, then one can write f 1 f 2 f n for their composition. 3.3.2 Inverse Functions Definition 3.3.3 Let A and B be nonempty sets and f F(A, B). Then f is invertible if there is a function f 1 F(B, A) such that and f f 1 = i B f 1 f = i A. If f 1 exists, it is called the inverse of f. If f is invertible, then f 1 is invertible and ( f 1 ) 1 = f Proposition 3.3.4 There is only one function that can be the inverse of a function f. Proof: Let A be a set and i A : A A be the identity function. Then i A is invertible and i 1 A = i A. absmath.tex; June 11, 2007; 15:24; p. 41

40 CHAPTER 3. FUNCTIONS Examples. Theorem 3.3.1 Let A and B be sets, and let f F(A, B). Then f is invertible if and only if f is a bijection. Proof: Examples. 3.3.3 Homework Reading: Sect 3.3 Exercises: 3.3[1,20] absmath.tex; June 11, 2007; 15:24; p. 42

Chapter 4 Binary Operations and Relations 4.1 LECTURE 11. Binary Operations Example Definition 4.1.1 function Let A be a nonempty set. A binary operation is a f : A A A. Notation. f (a, b), a b, a b, a + b, a b, a b, The element f (a, b) must be in the set A. Examples. 4.1.1 Associate and Commutative Laws Definition 4.1.2 A binary operation on A is associative if (a b) c = a (b c), a, b, c A Definition 4.1.3 A binary operation on A is commutative if a b = b a, a, b A 41

42 CHAPTER 4. BINARY OPERATIONS AND RELATIONS To prove that an operation is associative or commutative one needs to prove the property for all elements of A. To prove that an operation is not associative or commutative one needs to show that the property does not hold for some elements of A. Examples. Union of sets on P(U) Intersection of sets on P(U) Composition of functions on F(A) Multiplication and addition on F(R) Matrix multiplication on M 2 (R) Matrix addition on M 2 (R) 4.1.2 Identities Definition 4.1.4 Let be a binary operation on the set A. An element e is an identity element of A with respect to if for any a A. a e = e a = a Not all sets have identity elements with respect to a given binary operation. Examples. Proposition 4.1.1 If e is an identity element of the set A with respect to a binary operation on A, then e is unique. Proof: By contradiction. absmath.tex; June 11, 2007; 15:24; p. 43

4.1. LECTURE 11. BINARY OPERATIONS 43 4.1.3 Inverses Definition 4.1.5 Let be a binary operation on the set A with identity e. We say that a A is invertible with respect to if there exists b A such that If a b = b a = e. a b = b a = e, then b is an inverse of a with respect to. Examples. Proposition 4.1.2 Let be an associative binary operation on a set A with an identity element e. If a A has an inverse with respect to, then that inverse is unique. Proof: Exercise. Notation. The inverse of a is denoted by a 1. 4.1.4 Closure Definition 4.1.6 Let A be a nonempty set and X A. Let be a binary operation on the set A. If for any x, y X, we have x y X, then X is closed in A under. If X A is closed in A under, then is also a binary operation on X. Examples. To prove that X A is not closed under a binary operation, we need to prove that there exist x, y X such that x y X. Examples. absmath.tex; June 11, 2007; 15:24; p. 44

44 CHAPTER 4. BINARY OPERATIONS AND RELATIONS 4.1.5 Groups Definition 4.1.7 A set with a binary operation is a group if 1. the binary operation is associative, 2. there is an identity element, 3. every element has an inverse. Examples. 4.1.6 Homework Reading: Sect 4.1 Exercises: 4.1[2,5,19,22,36,37] absmath.tex; June 11, 2007; 15:24; p. 45

4.2. LECTURE 12. EQUIVALENCE RELATIONS 45 4.2 LECTURE 12. Equivalence Relations 4.2.1 Relations Definition 4.2.1 A relation R on a set A is a subset of A A. Notation. If (a, b) R, then we write arb Examples. 4.2.2 Properties of Relations The relation R = {(a, a) a A} is called the diagonal of A A. Definition 4.2.2 Let R be a relation on a set A. Then 1. R is reflexive if a A, ara, 2. R is symmetric if a, b A, if arb, then bra, 3. R is transitive if a, b, c A, if arb and brc, then arc, 4. R is antisymmetric if a, b A, if arb and bra, then a = b. Examples. absmath.tex; June 11, 2007; 15:24; p. 46

46 CHAPTER 4. BINARY OPERATIONS AND RELATIONS 4.2.3 Equivalence Relations Definition 4.2.3 it is A relation R on a set A is an equivalence relation if 1. reflexive, 2. symmetric and 3. transitive. Notation. An equivalence relation is denoted by a b read as a is equivalent to b Examples. Congruence mod n is a relation R on Z + defined by arb if a b = nk for some k Z written a b(mod n) Example 4.2.4 Equivalence Classes Definition 4.2.4 any a A the set Let be an equivalence relation on a set A. Then for [a] = {x A x a} is the equivalence class of a. Equivalence class of a is the set of all elements equivalent to a. absmath.tex; June 11, 2007; 15:24; p. 47

4.2. LECTURE 12. EQUIVALENCE RELATIONS 47 All elements of an equivalence class are equivalent to each other. Examples. An equivalence relation divides the set into disjoint equivalence classes. Theorem 4.2.1 The set of equivalence classes of an equivalence relation on a nonempty set A forms a partition of A. Proof: 1. Equivalence classes are subsets of A. 2. Equivalence classes are nonempty. 3. Every element of A is in some equivalence class. 4. Different equivalence classes are disjoint. In other words, or if [a] [b], then [a] [b] =, if [a] [b], then a b and [a] = [b]. Theorem 4.2.2 Let P be a partition of a nonempty set A. Let R be a relation on A defined by arb if a and b are in the same element of the partition. Then R is an equivalence relation. Proof: 1. R is reflexive. 2. R is symmetric. 3. R is transitive. There is a bijection between the set of all equivalence relations of A and the set of all partitions of A. absmath.tex; June 11, 2007; 15:24; p. 48

48 CHAPTER 4. BINARY OPERATIONS AND RELATIONS 4.2.5 Partial and Linear Ordering Example. Ordering of the real numbers. arb if a b R is reflexive, transitive and antisymmetric. Definition 4.2.5 R is A relation R on a set A is a partial ordering on A if 1. reflexive, 2. transitive and 3. antisymmetric. A set with a partial ordering is a partially ordered set. The relation < is not a partial ordering on R. Let A be a set and P(A) be the power set of A. Let R be a relation on P(A) defined by Then R is a partial ordering on P(A). XRY if X Y. Example. Linear ordering of real numbers. x, y R, either x y or y x. Definition 4.2.6 Let A be a set and R be a partial ordering on A. Then R is a linear ordering on A if a, b A, either arb or bra. A set with a linear ordering is a linearly ordered set. Examples. absmath.tex; June 11, 2007; 15:24; p. 49

4.2. LECTURE 12. EQUIVALENCE RELATIONS 49 4.2.6 Homework Reading: Sect 4.2 Exercises: 4.2[10,14,15,17] absmath.tex; June 11, 2007; 15:24; p. 50

50 CHAPTER 4. BINARY OPERATIONS AND RELATIONS absmath.tex; June 11, 2007; 15:24; p. 51

Chapter 5 The Integers 5.1 LECTURE 13. Axioms and Basic Properties 5.1.1 The Axioms of the Integers Definition 5.1.1 The set Z of integers is a set with two binary operations: addition, +, and multiplication,, with the properties A1 Associativity of addition: (x + y) + z = x + (y + z), x, y, z Z A2 Commutativity of addition: x + y = y + x, x, y Z A3 Additive identity: 0 Z such that x + 0 = x, x Z A4 Additive inverse: x Z, ( x) Z such that x + ( x) = 0 51

52 CHAPTER 5. THE INTEGERS Definition 5.1.2 A5 Associativity of multiplication: (x y) z = x (y z), x, y, z Z A6 Commutativity of multiplication: A7 Multiplicative identity: A8 Distributivity: Notation. x y = y x, x, y Z 1 Z such that x 1 = x, x Z, and 1 0 x (y + z) = x y + x z, x, y, z Z x + ( y) = x y, x y = xy These properties of integers are axioms. The multiplicative inverses of integers do not exist, in general. The axiom 1 0 is needed for a nontrivial theory. If 1 = 0, then x = 0 for any x Z. Therefore, Z = {0} consists of only one element. Proposition 5.1.1 Let a, b, c Z. Then P1 If a + b = a + c, then b = c P2 a0 = 0 Proof: P3 ( a)b = a( b) = (ab) P4 ( a) = a absmath.tex; June 11, 2007; 15:24; p. 52

5.1. LECTURE 13. AXIOMS AND BASIC PROPERTIES 53 Proposition 5.1.2 Let a, b, c Z. Then P5 ( a)( b) = ab P6 a(b c) = ab bc Proof: P7 ( 1)a = a P8 ( 1)( 1) = 1 Definition 5.1.3 The set of positive integers Z + is defined by Z + = {n Z n = 1 } + {{ + } 1} n Notation. Z + = N Definition 5.1.4 The set Z + has the properties A9 Closure. Z + is closed with respect to addition and multiplication: if x, y Z +, then x + y Z + and xy Z + A10 Trichotomy Law. For every integer x Z exactly one of the following statements is true: 1. x Z +, 2. x Z +, 3. x = 0. Proposition 5.1.3 If x Z and x 0, then x 2 Z +. Proof: Follows from closure of Z +. 5.1.2 Inequalities Definition 5.1.5 Let x, y Z. Then x < y if y x Z +. absmath.tex; June 11, 2007; 15:24; p. 53

54 CHAPTER 5. THE INTEGERS Notation. x < y is read x is less than y if x < y, then y > x if x < y or x = y, then x y if x > y or x = y, then x y if x y, then y x If x Z +, then x > 0. Proposition 5.1.4 Let a, b, c Z. Q1 Exactly one of the following holds: Q4 If a > 0 and b < 0, then ab < 0. a < b, b < a or b = a. Q2 If a > 0, then a < 0. (If a < 0, then a > 0.) Q3 If a > 0 and b > 0, then a + b > 0 and ab > 0. Q5 If a < 0 and b < 0, then ab > 0. Q6 If a < b and b < c, then a < c. Q7 If a < b, then a + c < b + c. Q8 If a < b and c > 0, then ac < bc. Q9 If a < b and c < 0, then ac > bc. Proof: Exercise. 5.1.3 The Well-Ordering Principle Definition 5.1.6 A11. The Well-Ordering Principle. Every nonempty subset of Z + has a smallest element. That is, if S is a nonempty subset of Z +, then there is a S such that a x for all x S. absmath.tex; June 11, 2007; 15:24; p. 54

5.1. LECTURE 13. AXIOMS AND BASIC PROPERTIES 55 Proposition 5.1.5 There is no integer x such that 0 < x < 1. Proof: 1. By contradiction. 2. Suppose that a Z + is the smallest integer such that 0 < a < 1. 3. Then a 2 < a (contradiction). Corollary 5.1.1 The number 1 is the smallest element of Z +. Proof: Exercise. Corollary 5.1.2 The only integers having multiplicative inverses in Z are 1 and 1. Proof: By contradiction. 5.1.4 Homework Reading: Sect 5.1 Exercises: 5.1[1,9] absmath.tex; June 11, 2007; 15:24; p. 55

56 CHAPTER 5. THE INTEGERS 5.2 LECTURE 14. Induction 5.2.1 Induction: A Method of Proof Induction is a method for proving statements about the positive integers. Let P(x) be an open sentence. The purpose of an induction proof is to show that the statement P(n) is true for every positive integer n Z +. Idea: 1. Verify (prove) that P(1) is true. 2. Given a positive integer k Z + for which P(k) is true, prove that P(k + 1) is true. 3. This establishes that P(n) is true for any n Z +. Theorem 5.2.1 First Principle of Mathematical Induction. Let P(n) be a statement about the positive integer n. Suppose that 1. P(1) is true, and 2. If P(k) is true for a k Z +, then P(k + 1) is true. Then P(n) is true for any n Z +. Proof: By contradiction and the Well-Ordering Principle. The assumption that P(k) is true is the induction hypothesis. Examples. 5.2.2 Other Forms of Induction Theorem 5.2.2 Modified Form of First Principle of Mathematical Induction Let P(n) be a statement about the integer n. Suppose that there is an integer n 0 Z such that 1. P(n 0 ) is true, and 2. If P(k) is true for an integer k n 0, then P(k + 1) is true. Then P(n) is true for any integer n n 0. absmath.tex; June 11, 2007; 15:24; p. 56

5.2. LECTURE 14. INDUCTION 57 Proof: Exercise. Example Theorem 5.2.3 Second Principle of Mathematical Induction. P(n) be a statement about the positive integer n. Suppose that Let 1. P(1) is true, and 2. If, for a positive integer k Z +, P(i) is true for every positive integer i k, then P(k + 1) is true. Then P(n) is true for any positive integer n Z +. Proof: Exercise. A function f : Z + Z + is defined recursively if the value f (k) of f at a positive integer k is defined by the values f (1), f (2),..., f (k 1) of f at the preceding positive integers. f (k) = F( f (1),..., f (k 1)) This equation is a recursion, or a recursive relation. Example. Let f : Z + Z + be defined by f (1) = 1, f (2) = 5, and Then Proof: By induction. f (n + 1) = f (n) + 2 f (n 1). f (n) = 2 n + ( 1) n Restatement of the First Principle of Mathematical Induction in the language of the set theory. Theorem 5.2.4 that Let S Z + be a subset of positive integers. Suppose 1. 1 S, and 2. If k S, then k + 1 S. Then S = Z +. absmath.tex; June 11, 2007; 15:24; p. 57

58 CHAPTER 5. THE INTEGERS Proof: Exercise. Example n k = k=1 n(n + 1) 2 Theorem 5.2.5 Let A be a set and {B i } n i=1 be a finite collection of sets. Then n n A B i = ( ) A B i Proof: By induction. 5.2.3 The Binomial Theorem i=1 i=1 Definition 5.2.1 Let n be a nonnegative integer. The factorial n! of n is defined by and 0! = 1, n! = n(n 1) 2 1 if n > 0. Definition 5.2.2 Let n and r be nonnegative integers such that 0 r n. The binomial coefficient ( n r) is defined by ( ) n n! n(n 1) (n r + 1) = = r r!(n r)! r! Notation. ( ) n = Cr n r Note that ( ) 0 = 0 Also ( ) n = n ( ) n = 1 0 ( ) ( ) n n = k n k absmath.tex; June 11, 2007; 15:24; p. 58

5.2. LECTURE 14. INDUCTION 59 The binomial coefficient ( n r) is the number of ways to choose r objects from a collection of n objects. The binomial coefficient ( n r) is the number of subsets of r elements in a set with n elements. Identity. Let n, k Z + such that 1 k n. Then ( ) ( ) ( ) n + 1 n n = + k k k 1 Proof: Exercise. Theorem 5.2.6 Let a, b Z and n Z +. Then n ( ) n (a + b) n = a n k b k k Proof: By induction. Example Corollary 5.2.1 Let n Z be a nonnegative integer, n 0. Then n k=0 k=0 ( ) n = 2 n k Proof: Apply the binomial theorem to (1 + 1) n. 5.2.4 Homework Reading: Sect 5.2 Exercises: 5.2[1,2,4,26] absmath.tex; June 11, 2007; 15:24; p. 59

60 CHAPTER 5. THE INTEGERS 5.3 LECTURE 15. The Division Algorithm 5.3.1 Divisors and Greatest Common Divisors Definition 5.3.1 Let a and b be integers. We say b divides a if there is an integer c such that a = bc. The integer a is divisible by b and c. The integers b and c are factors of a. Notation. Example Proposition 5.3.1 Let a, b, c Z. 1. If a 1, then a = 1 or a = 1. b a (read b divides a ) 2. If a b and b a, then a = b or a = b. 3. If a b and a c, then a (bx + cy) for any x, y Z. 4. If a b and b c, then a c. Proof: Exercise. Example Definition 5.3.2 Let a and b be integers, not both zero. A common divisor of a and b is an integer c such that c divides both a and b. A greatest common divisor of a and b is a positive integer d such that 1. d is a common divisor of a and b, and 2. for any integer c, if c is a common divisor of a and b, then c divides d. Notation. The greatest common divisor (gcd) of two integers a and b is denoted by absmath.tex; June 11, 2007; 15:24; p. 60

5.3. LECTURE 15. THE DIVISION ALGORITHM 61 Note that for any a Z, a 0, Example gcd(a, b) = (a, b) gcd(a, 0) = a Theorem 5.3.1 Division Algorithm. Let a be an integer and b be a positive integer. Then there exist unique integers q and r such that Proof: a = bq + r and 0 r < b. 1. Let S = {a bx x Z} and S 0 = {n S n 0}. 2. Show that S 0. (If a 0, then a S 0. If a < 0, then a ba S 0.) 3. Let r be the smallest element of S 0. Then r 0. 4. Also, r = a bq for some q Z. 5. Then r < b, since if r b, then r b < r is the smallest element of S 0 (contradiction). 6. Uniqueness. Assume q 1, r 1 and q 2, r 2 and show that q 1 = q 2 and r 1 = r 2. Theorem 5.3.2 Let a and b be integers, not both zero. Then: 1. the greatest common divisor d of a and b exists and is unique, 2. there exist integers x and y such that Proof: d = ax + by. 1. Let S = {ax + by x, y Z} and S 0 = {n S n > 0}. 2. Then ±a, ±b S. So, S 0. 3. Let d be the smallest element of S 0. absmath.tex; June 11, 2007; 15:24; p. 61

62 CHAPTER 5. THE INTEGERS 4. Then d S. Hence d = ax + by for some x, y Z. 5. Also, there exist q, r Z such that a = dq + r and 0 r < d. 6. Since r = a(1 xq) + b( yq), then r S. 7. It is impossible that r > 0 since then r S 0 and r < d (contradiction). Therefore, r = 0. 8. So, a = dq, or d a. Similarly, d b. 9. Let c Z be a common divisor of a and b. Then a = cu and b = cv for some u, v Z. 10. Therefore, d = ax + by = c(ux + vy). So, c d. 11. Thus d = gcd(a, b). 12. Uniqueness. Exercise. 5.3.2 Euclidean Algorithm Lemma 5.3.1 Let a and b be integers, not both zero. If there exist two integers q and r such that then Proof: a = bq + r and 0 r < b, gcd(a, b) = gcd(b, r) 1. Let d = gcd(a, b). 2. Then d r. 3. For any c Z, if c b and c r, then c a. 4. Since d = ax + by, then c d. Euclidean Algorithm is the following procedure for finding the gcd of two integers. absmath.tex; June 11, 2007; 15:24; p. 62