Chapter 1 : The language of mathematics.

Size: px
Start display at page:

Transcription

1 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 : P Q P or Q T T T T F T F T T F F F Truth table for the connective and : P Q P and Q T T T T F F F T F F F F Truth table for the connective not : P T F not P F T Chapter 2 : Implications. Truth table for the implication : P Q P Q T T T T F F F T T F F T Chapter 3 : Proofs. Trichotomy law : For real numbers a and b, one and only one of the three possibilities a < b, a = b, a > b is true. 1

2 2 Addition law : For real numbers a, b and c, a < b a + c < b + c. Multiplication law : For real numbers a, b and c, a < b ac < bc if c > 0; a < b ac > bc if c < 0. Transitive law : For real numbers a, b and c, a < b and b < c a < c. Chapter 4 : Proof by contradiction. Consists of proving the statement (not P ) Q where Q is a false statement. Proving an implication P Q by contradiction : assume P is true and Q is false, and obtain a contradiction. Proof by contrapositive : based on the fact that the statements P Q and its contrapositive (not Q) (not P ) are logically equivalent. Proving or statements : based on the fact that the statements P or Q and (not P ) Q are logically equivalent. Proposition. Given non-zero real numbers a and b, we have ab > 0 if and only if a and b have the same sign, and ab < 0 if and only if a and b have opposite signs. Chapter 5 : The Induction Principle. Axiom (The induction principle). Suppose that P (n) is a statement involving a general positive integer n. Then P (n) is true for all positive integers 1, 2, 3,... if (i) P (1) is true, (ii) P (k) P (k + 1) for all positive integers k. Chapter 6 : The language of set theory. Definition. A set is any well-defined collection of objects. Examples : Z is the set of all integers.

3 Z is the set of all non-negative integers 0, 1, 2, 3,... N is the set of all positive integers 1, 2, 3,... Q is the set of all rational numbers (fractions). R is the set of all real numbers. Definition. The objects in a set are called elements, members or points. We write x E to denote the fact that the object x is an element of the set E. Specifying a set : Listing the elements, example : A = {1, 3, π, 5}. The order is not important and there is no repetition. Conditional definition, example : A = {n N : 1 < n < 8}. Constructive definition, example : A = {n 2 +1 : n N} = {2, 5, 10, 17,... }. Definition. Two sets A and B are equal, written A = B, if they have precisely the same elements, i.e. x A x B. Definition. The empty set is the unique set which has no elements at all. It is denoted by. Definition. Given sets A and B, we say that A is a subset of B, written A B, if every element of A is an element of B, i.e. x A x B. In this case, if in addition A is not equal to B (so that B contains an element which is not an element of A), then we say that A is a proper subset of B and write A B. Thus A = B if and only if A B and B A. Operations on sets : Intersection : A B = {x : x A and x B}. Union : A B = {x : x A or x B}. Difference : A \ B = {x : x A and x / B}. Definition. Two sets A and B are said to be disjoint if A B =. Definition. The power set of a set X, denoted by P(X), is the set of all subsets of X. Thus A P(X) if and only if A X. Definition. Once we have fixed a universal set U, we can define the complement of any A P(U), denoted by A c, by A c = U \ A. Theorem. Let A, B, C be subsets of some universal set U. Then (i) Associativity : A (B C) = (A B) C, A (B C) = (A B) C; (ii) Commutativity : A B = B A, A B = B A; (iii) Distributivity : A (B C) = (A B) (A C), A (B C) = (A B) (A C); (iv) De Morgan laws : (A B) c = A c B c, (A B) c = A c B c ; 3

4 4 (v) Complementation : A A c = U, A A c = ; (vi) Double complements : (A c ) c = A. Chapter 7 : Quantifiers. Universal statement : a A, P (a) means that for all elements a in the set A, the proposition P (a) is true. Example : n N, n > 0. Existential statement : a A, P (a) means that there exists an element a in the set A for which the proposition P (a) is true. Example : x R, x 2 = π. Proving and disproving statements involving quantifiers : Proving statements of the form a A, P (a) : it suffices to prove the implication a A P (a). Proving statements of the form a A, P (a) : it suffices to exhibit a particular element a A for which P (a) is true. Disproving statements of the form a A, P (a) : we prove the negation a A, not P (a) by giving a counterexample, i.e. an element in A for which P (a) is false. Disproving statements of the form a A, P (a) : we prove the negation a A, not P (a). Definition. Given sets X and Y, the Cartesian product of X and Y, denoted by X Y, is the set of all ordered pairs (x, y) where x X and y Y X Y = {(x, y) : x X, y Y }. When Y = X, we write X X = X 2. Chapter 8 : Functions. Definition. Let X and Y be sets. A function f : X Y is the assignment of a unique element of Y to each element of X. We denote the element of Y assigned to x X by f(x); it is called the value of f at x or the image of x under f. The set X is called the domain of the function f and the set Y is called the codomain. Definition. The identity function I X : X X is defined by I X (x) = x for all x X. Definition. Two functions f : X Y and g : X Y are equal, written f = g, if they have the same value at each point of the domain X, i.e. f(x) = g(x) for all x X.

5 Definition. Given two functions f : X Y and g : Y Z, the composite of f and g, denoted by g f : X Z, is the function defined by (g f)(x) = g(f(x)) (x X). Proposition. Suppose that f : X Y, g : Y Z and h : Z W are functions. Then (i) (h g) f = h (g f). (ii) f I X = f = I Y f. Definition. Given a function f : X Y, we define its image by Im f = {f(x) : x X}. It is the subset of the codomain Y consisting of all values of f. Definition. Suppose that f : X Y is a function. Then we define the graph of f to be the subset of X Y given by G f = {(x, f(x)) : x X}. 5 Chapter 9 : Injections, surjections and bijections. Definition. Suppose that f : X Y is a function. (i) We say that f is injective if x 1, x 2 X, (f(x 1 ) = f(x 2 ) x 1 = x 2 ); (ii) We say that f is surjective if y Y, x X, y = f(x); (iii) We say that f is bijective if it is both injective and surjective. Definition. For a function f : X Y and an element y Y, we say that an element x X is a preimage of y under f if y = f(x). Thus f is injective if and only if every element of Y has at most one preimage. f is surjective if and only if every element of Y as at least one preimage. f is bijective if and only if every element of Y has exactly one preimage. Definition. A function f : X Y is invertible if there exists a function g : Y X such that y = f(x) x = g(y) ( x X, y Y ). In this case, we say that g is the inverse of f and write g = f 1. The symmetry of the definition shows that in this case g is also invertible and f is the inverse of g.

6 6 Theorem. Let f : X Y be a function. Then f is invertible if and only if it is bijective. Furthermore, if this is the case, then the inverse of f is unique. Proposition. The functions f : X Y and g : Y X are inverses to each other if and only if g f = I X and f g = I Y. Definition. Let f : X Y be a function. (i) The function f : P(X) P(Y ) is defined by f(a) = {f(x) : x A} for A P(X); (ii) The function f 1 : P(Y ) P(X) is defined by f 1 (B) = {x X : f(x) B}. Chapter 10 : Counting. Given n N, we define N n = {1, 2,..., n}. Definition. Let X be a set. If there is a bijection f : N n X, then we say that the cardinality of X is n and write X = n. The cardinality of the empty set is defined to be 0. Proposition. Suppose that f : N m X and g : N n X are bijections. Them m = n. Definition. Let X be a set. If X = n for some non-negative integer n, then we say that X is finite. Otherwise, we say that X is infinite. Theorem (Addition principle). Suppose that X and Y are disjoint finite sets. Then X Y is finite and X Y = X + Y. Corollary. Suppose that X 1, X 2,..., X n are pairwise disjoint finite sets. Then X 1 X 2 X n is finite and X 1 X 2 X n = X 1 + X X n. Theorem (Multiplication principle). Suppose that X and Y are finite sets. Then X Y is finite and X Y = X Y. Proposition (Inclusion-exclusion principle for two sets). Suppose that X and Y are finite sets. Then X Y is finite and X Y = X + Y X Y.

7 Proposition (Inclusion-exclusion principle for three sets). Suppose that X, Y and Z are finite sets. Then X Y Z is finite and X Y Z = X + Y + Z X Y X Z Y Z + X Y Z. 7 Chapter 11 : Properties of finite sets. Theorem (Pigeonhole principle). Let X and Y be non-empty finite sets. If there exists an injection X Y, then X Y. Equivalently, if X > Y, then every function f : X Y is not injective. Theorem. Let X and Y be non-empty finite sets. If there exists a surjection X Y, then X Y. Equivalently, if X < Y, then every function f : X Y is not surjective. Proposition. Suppose that X Y where Y is a finite set. Then X is also finite and X Y. Theorem. Let X and Y be two non-empty finite sets and suppose that X = Y. Then a function f : X Y is injective if and only if it is surjective. Definition. Let A R. We say that b is a minimal element of A, written b = min A, if (i) b A; (ii) a A b a. Similarly, we say that c is a maximal element of A, written c = max A, if (i) c A; (ii) a A c a. The maximal element and minimal element, if they exist, are unique. Proposition. Let A be a finite non-empty set of real numbers. Then A has a minimum element and a maximum element. Chapter 12 : Counting functions and subsets. Proposition (Number of functions). Suppose that X and Y are non-empty finite sets with X = m and Y = n. Then Fun(X, Y ) = n m. Proposition (Number of injective functions). Let X and Y be non-empty finite sets with X = m and Y = n. Suppose that m n. Then Inj(X, Y ) = n! (n m)!.

8 8 Definition. Given a set X, a bijection X X is called a permutation of the set X. Proposition (Number of permutations). Suppose that X is a finite nonempty set of cardinality n. Then the number of permutations of X is n!. Proposition (Number of subsets). Suppose that X is a finite non-empty set. Then P(X) = 2 X. Definition. Given a set X and a non-negative integer r, an r-subset of X is a subset A X of cardinality r. We denote by P r (X) the set of all r-subsets of X : P r (X) = {A X : A = r}. We define the binomial coefficient ( n r) to be the cardinality of Pr (X) when X = n. Proposition. For n and r non-negative integers, we have (i) ( n r) = 0 if r > n; (ii) ( ( n 0) = 1, n ( 1) = n, n n) = 1; (iii) ( ) ( n r = n n r) for 0 r n. Proposition. n j=0 ( ) n = 2 n. j Proposition (Pascal s law). For positive integers n and r such that 1 r n, we have ( ) ( ) ( ) n n 1 n 1 = +. r r 1 r Theorem. For non-negative integers n and r such that r n, we have ( ) n n! = r r!(n r)!. Theorem (Binomial theorem). For all real numbers a and b and nonnegative integers n, we have n ( ) n (a + b) n = a n j b j. j j=0 Chapter 13 : Number systems. Theorem (Irrationality of 2). There does not exist a rational number whose square is 2.

9 9 Chapter 14 : Counting infinite sets. Definition. Two sets X and Y are equipotent if there is a bijection X Y. Definition. A set X is said to be denumerable if there is a bijection N X. A set is countable if it is either finite or denumerable. A set is uncountable if it is not countable. If X is denumerable, then we say that its cardinality is ℵ 0 and we write X = ℵ 0. Examples of denumerable sets : Z. Z. the set of even integers. Proposition. Let X and Y be sets and suppose that X is denumerable. Then Y is also denumerable if and only if it is equipotent to X. Proposition. If A and B are denumerable, then so is their union A B and their Cartesian product A B. Proposition. A subset of a denumerable set is countable. Theorem (Denumerability of the rationals). The set of rational numbers Q is denumerable. Theorem (Uncountability of the reals). The set of real numbers R is uncountable. Definition. We say that two (possibly infinite) sets X and Y have the same cardinality, written X = Y, if they are equipotent. If there is an injection X Y, then we write X Y. We write X < Y to mean that X Y but X Y. Theorem. For any set X, we have X < P(X). Theorem (Cantor Schröder Bernstein theorem). Suppose that X and Y are non-empty sets with X > Y. Then any function f : X Y is not injective. Corollary. Let X and Y be non-empty sets. If X Y and Y X, then X = Y. Definition. A real number x is called algebraic if it satisfies a polynomial equation a 0 + a 1 x + + a n x n = 0, where the coefficients a 0, a 1,..., a n are integers. If x is not algebraic, then we say that it is transcendental.

10 10 Examples of algebraic numbers : any rational number. the square root of any integer. Proposition. The set of algebraic numbers is denumerable. Chapter 15 : The division theorem. Theorem. Let a, b be integers with b > 0. Then there are unique integers q and r such that a = qb + r and 0 r < b. The integer q is called the quotient and r is called the remainder. Note that b divides a if and only if r = 0. Chapter 16 : The Euclidean algorithm. Definition. Let a and b be two integers, not both zero. The greatest common divisor of a and b, noted (a, b), is the unique positive integer d such that (i) d divides b and d divides a; (ii) If c divides a and c divides b, then c d. The Euclidean algorithm is a procedure which gives the greatest common divisor of two integers a and b. It is based on the following lemmas. Lemma. If a positive integer b divides a, then (a, b) = b. Lemma. Let a and b be non-zero integers and suppose that a = qb + r for some q, r Z. Then (a, b) = (b, r). Chapter 17 : Consequences of the Euclidean algorithm. Theorem. Let a and b be integers not both zero. Then there exist integers m and n such that ma + nb = (a, b). The integers m, n can be obtained by going backwards in the Euclidean algorithm. Corollary. Let a and b be integers not both zero. divisor of a and b if and only if c divides (a, b). Then c is a common

11 Definition. Two integers a and b not both zero are called coprime if (a, b) = 1, in other words their only common divisors are 1 and 1. Proposition. Let a and b be two integers not both zero. Then a and b are coprime if and only if there exist integers m and n such that ma + nb = 1. Theorem. Let a, b and c be positive integers with a and b coprime. If a divides bc, then a divides c. 11 Chapter 18 : Linear Diophantine equations. Theorem. For positive integers a, b and c, there exist integers m and n such that ma + nb = c if and only if (a, b) divides c. Theorem. Let a, b, c be positive integers and suppose that (a, b) divides c. Let m 0, n 0 be integers such that Then m, n is another solution to if and only if and for some q Z. m 0 a + n 0 b = c. ma + nb = c m = m 0 + n = n 0 b (a, b) q a (a, b) q Chapter 19 : Congruence of integers. Definition. Let m be an integer greater than one. We say that the integers a and b are congruent modulo m if m divides a b. In this case, we write a b mod m. Proposition. (i) Reflexive property : For all integers a, a a mod m; (ii) Symmetric property : If a and b are integers such that a b mod m, then b a mod m; (iii) Transitive property : If a, b and c are integers such that a b mod m and b c mod m, then a c mod m.

12 12 Proposition (Modular arithmetic). Suppose that a 1, a 2, b 1, b 2 are integers such that a 1 a 2 mod m and b 1 b 2 mod m. Then (i) a 1 + b 1 a 2 + b 2 mod m; (ii) a 1 b 1 a 2 b 2 mod m; (iii) a 1 b 1 a 2 b 2 mod m. Definition. The set of integers R m = {0, 1, 2,..., m 1} is called the set of residues modulo m. Proposition. Given an integer a, there is a unique r R m such that a r mod m. Proposition (Division in congruences). We have ab 1 ab 2 mod m b 1 b 2 mod m/(a, m). In particular, if a divides m, then ab 1 ab 2 mod m b 1 b 2 mod m/a and if a and m are coprime, then ab 1 ab 2 mod m b 1 b 2 mod m. Chapter 23 : The sequence of prime numbers. Definition. A positive integer n is said to be prime if n > 1 and the only positive divisors of n are 1 and n. If an integer n > 1 is not prime, then it is said to be composite. Proposition. Every integer greater than 1 can be written as a product of prime numbers. Theorem. Let a, b be positive integers and let p be a prime number. If p divides ab, then p divides a or p divides b. Proposition. If n 2 is a composite number, then it has a prime divisor not exceeding n. Theorem (Fundamental theorem of arithmetic). Every positive integer greater than 1 can be written uniquely as a product of prime numbers with the prime factors in the product written in non-decreasing order. Theorem. There are infinitely many prime numbers.

586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 465 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

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,

Math 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,

Final 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,

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3

Part IA Numbers and Sets

Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

Introduction 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

Discrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland

Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................

Sets. We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth

Sets We discuss an informal (naive) set theory as needed in Computer Science. It was introduced by G. Cantor in the second half of the nineteenth century. Most students have seen sets before. This is intended

MATH 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

Copyright 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

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

Name (please print) Mathematics Final Examination December 14, 2005 I. (4)

Mathematics 513-00 Final Examination December 14, 005 I Use a direct argument to prove the following implication: The product of two odd integers is odd Let m and n be two odd integers Since they are odd,

MATH 363: Discrete Mathematics

MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The

Section 0. Sets and Relations

0. Sets and Relations 1 Section 0. Sets and Relations NOTE. Mathematics is the study of ideas, not of numbers!!! The idea from modern algebra which is the focus of most of this class is that of a group

SETS 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,

Contribution 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

5 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,

FOUNDATIONS & PROOF LECTURE NOTES by Dr Lynne Walling

FOUNDATIONS & PROOF LECTURE NOTES by Dr Lynne Walling Note: You are expected to spend 3-4 hours per week working on this course outside of the lectures and tutorials. In this time you are expected to review

Prof. 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

Foundations 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

Part IA Numbers and Sets

Part IA Numbers and Sets Theorems Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

Notation Index. gcd(a, b) (greatest common divisor) NT-16

Notation Index (for all) B A (all functions) B A = B A (all functions) SF-18 (n) k (falling factorial) SF-9 a R b (binary relation) C(n,k) = n! k! (n k)! (binomial coefficient) SF-9 n! (n factorial) SF-9

MATH 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

1.4 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

In 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

In 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

RED. Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam

RED Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam Note that the first 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choice

Foundations Revision Notes

oundations Revision Notes hese notes are designed as an aid not a substitute for revision. A lot of proofs have not been included because you should have them in your notes, should you need them. Also,

MATH 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

Discrete Mathematics

Discrete Mathematics Dr. Thomas Baird January 7, 2013 Contents 1 Logic 2 1.1 Statements.................................... 2 1.1.1 And, Or, Not.............................. 2 1.1.2 Implication...............................

MAT115A-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

Discrete Math Notes. Contents. William Farmer. April 8, Overview 3

April 8, 2014 Contents 1 Overview 3 2 Principles of Counting 3 2.1 Pigeon-Hole Principle........................ 3 2.2 Permutations and Combinations.................. 3 2.3 Binomial Coefficients.........................

MATH 2200 Final Review

MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics

Propositional 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

0 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

Exercises 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

MATH 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

Discussion Summary 10/16/2018

Discussion Summary 10/16/018 1 Quiz 4 1.1 Q1 Let r R and r > 1. Prove the following by induction for every n N, assuming that 0 N as in the book. r 1 + r + r 3 + + r n = rn+1 r r 1 Proof. Let S n = Σ n

A Semester Course in Basic Abstract Algebra

A Semester Course in Basic Abstract Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved December 29, 2011 1 PREFACE This book is an introduction to abstract algebra course for undergraduates

1. (B) The union of sets A and B is the set whose elements belong to at least one of A

1. (B) The union of sets A and B is the set whose elements belong to at least one of A or B. Thus, A B = { 2, 1, 0, 1, 2, 5}. 2. (A) The intersection of sets A and B is the set whose elements belong to

MATH1240 Definitions and Theorems

MATH1240 Definitions and Theorems 1 Fundamental Principles of Counting For an integer n 0, n factorial (denoted n!) is defined by 0! = 1, n! = (n)(n 1)(n 2) (3)(2)(1), for n 1. Given a collection of n

6 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

There are seven questions, of varying point-value. Each question is worth the indicated number of points.

Final Exam MAT 200 Solution Guide There are seven questions, of varying point-value. Each question is worth the indicated number of points. 1. (15 points) If X is uncountable and A X is countable, prove

Foundations 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

Functions 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

Analysis I. Classroom Notes. H.-D. Alber

Analysis I Classroom Notes H-D Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21

Equivalence of Propositions

Equivalence of Propositions 1. Truth tables: two same columns 2. Sequence of logical equivalences: from one to the other using equivalence laws 1 Equivalence laws Table 6 & 7 in 1.2, some often used: Associative:

Contribution 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

Topology. 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

Sets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions

Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Notation x A means that element x is a member of set A. x / A means that x is not a member of A.

INFINITY: CARDINAL NUMBERS

INFINITY: CARDINAL NUMBERS BJORN POONEN 1 Some terminology of set theory N := {0, 1, 2, 3, } Z := {, 2, 1, 0, 1, 2, } Q := the set of rational numbers R := the set of real numbers C := the set of complex

Contents Propositional Logic: Proofs from Axioms and Inference Rules

Contents 1 Propositional Logic: Proofs from Axioms and Inference Rules... 1 1.1 Introduction... 1 1.1.1 An Example Demonstrating the Use of Logic in Real Life... 2 1.2 The Pure Propositional Calculus...

CS Discrete Mathematics Dr. D. Manivannan (Mani)

CS 275 - Discrete Mathematics Dr. D. Manivannan (Mani) Department of Computer Science University of Kentucky Lexington, KY 40506 Course Website: www.cs.uky.edu/~manivann/cs275 Notes based on Discrete Mathematics

Background for Discrete Mathematics

Background for Discrete Mathematics Huck Bennett Northwestern University These notes give a terse summary of basic notation and definitions related to three topics in discrete mathematics: logic, sets,

MATH 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

Chapter Summary. Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.

Chapter 2 Chapter Summary Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.6) Section 2.1 Section Summary Definition of sets Describing

Review Problems for Midterm Exam II MTH 299 Spring n(n + 1) 2. = 1. So assume there is some k 1 for which

Review Problems for Midterm Exam II MTH 99 Spring 014 1. Use induction to prove that for all n N. 1 + 3 + + + n(n + 1) = n(n + 1)(n + ) Solution: This statement is obviously true for n = 1 since 1()(3)

Mathematics 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,...

MAT 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

Functions. Definition 1 Let A and B be sets. A relation between A and B is any subset of A B.

Chapter 4 Functions Definition 1 Let A and B be sets. A relation between A and B is any subset of A B. Definition 2 Let A and B be sets. A function from A to B is a relation f between A and B such that

Lecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel

Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical

MATH 2200 Final LC Review

MATH 2200 Final LC Review Thomas Goller April 25, 2013 1 Final LC Format The final learning celebration will consist of 12-15 claims to be proven or disproven. It will take place on Wednesday, May 1, from

Lecture 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...........................

Chapter Summary. Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability

Chapter 2 1 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability Sequences and Summations Types of Sequences Summation

Math 3000 Section 003 Intro to Abstract Math Final Exam

Math 3000 Section 003 Intro to Abstract Math Final Exam Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Name: Problem 1a-j 2 3a-b 4a-b 5a-c 6a-c 7a-b 8a-j

Math 105A HW 1 Solutions

Sect. 1.1.3: # 2, 3 (Page 7-8 Math 105A HW 1 Solutions 2(a ( Statement: Each positive integers has a unique prime factorization. n N: n = 1 or ( R N, p 1,..., p R P such that n = p 1 p R and ( n, R, S

Week 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

MATH 215 Final. M4. For all a, b in Z, a b = b a.

MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on

MATH 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

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

LECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel

LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is

Chapter 0. Introduction: Prerequisites and Preliminaries

Chapter 0. Sections 0.1 to 0.5 1 Chapter 0. Introduction: Prerequisites and Preliminaries Note. The content of Sections 0.1 through 0.6 should be very familiar to you. However, in order to keep these notes

7.11 A proof involving composition Variation in terminology... 88

Contents Preface xi 1 Math review 1 1.1 Some sets............................. 1 1.2 Pairs of reals........................... 3 1.3 Exponentials and logs...................... 4 1.4 Some handy functions......................

POL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005

POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1

MATH 3330 ABSTRACT ALGEBRA SPRING Definition. A statement is a declarative sentence that is either true or false.

MATH 3330 ABSTRACT ALGEBRA SPRING 2014 TANYA CHEN Dr. Gordon Heier Tuesday January 14, 2014 The Basics of Logic (Appendix) Definition. A statement is a declarative sentence that is either true or false.

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same

Introduction to Abstract Mathematics

Introduction to Abstract Mathematics Notation: Z + or Z >0 denotes the set {1, 2, 3,...} of positive integers, Z 0 is the set {0, 1, 2,...} of nonnegative integers, Z is the set {..., 1, 0, 1, 2,...} of

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.

The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring

Mathematical Reasoning & Proofs

Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0

Sets McGraw-Hill Education

Sets A set is an unordered collection of objects. The objects in a set are called the elements, or members of the set. A set is said to contain its elements. The notation a A denotes that a is an element

Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop Spis treści

Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop. 2017 Spis treści Preface xiii I Fundamentals of Abstract Mathematics 1 1 Basic Notions 3 1.1 A

Axioms for Set Theory

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:

Mathematics Review for Business PhD Students

Mathematics Review for Business PhD Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

REVIEW QUESTIONS. Chapter 1: Foundations: Sets, Logic, and Algorithms

REVIEW QUESTIONS Chapter 1: Foundations: Sets, Logic, and Algorithms 1. Why can t a Venn diagram be used to prove a statement about sets? 2. Suppose S is a set with n elements. Explain why the power set

2.1 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

Part 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.]

HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 1. I. Foundational material

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 1 Fall 2014 I. Foundational material I.1 : Basic set theory Problems from Munkres, 9, p. 64 2. (a (c For each of the first three parts, choose a 1 1 correspondence

MATH 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.

Packet #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics

CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus

Sets 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

9 FUNCTIONS. 9.1 The Definition of Function. c Dr Oksana Shatalov, Fall

c Dr Oksana Shatalov, Fall 2018 1 9 FUNCTIONS 9.1 The Definition of Function DEFINITION 1. Let X and Y be nonempty sets. A function f from the set X to the set Y is a correspondence that assigns to each

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 1. (a) Write the converse, contrapositive and negation of the following statement: For every integer n, if n is divisible by 3 then

Mathematics Review for Business PhD Students Lecture Notes

Mathematics Review for Business PhD Students Lecture Notes Anthony M. Marino Department of Finance and Business Economics Marshall School of Business University of Southern California Los Angeles, CA 90089-0804

Analysis 1. Lecture Notes 2013/2014. The original version of these Notes was written by. Vitali Liskevich

Analysis 1 Lecture Notes 2013/2014 The original version of these Notes was written by Vitali Liskevich followed by minor adjustments by many Successors, and presently taught by Misha Rudnev University

Theorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)

Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +

Part IA. Numbers and Sets. Year

Part IA Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2017 19 Paper 4, Section I 1D (a) Show that for all positive integers z and n, either z 2n 0 (mod 3) or