Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.


 Darrell Wood
 1 years ago
 Views:
Transcription
1 Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square of an integer} (c) {2,{2}} (d) {{2},{{2}}} (e) {{2},{2,{2}}} (f) {{{2}}} (a) Since the set contains only integers and {2} is a set, not an integer, {2} is not an element. (b) Since the set contains only integers and {2} is a set, not an integer, {2} is not an element. (c) The set has two elements. One of them is patently {2}. (d) The set has two elements. One of them is patently {2}. (e) The set has two elements. One of them is patently {2}. (f) The set has only one element, {{{2}}}; Since this is not the same as {2} (the former is a set containing a set, whereas the latter is a set containing a nubmer), {2} is not an element of {{{2}}}. 2. (8%) What is the cardinality of each of these sets? (a) ϕ (b) {ϕ} (c) {ϕ, {ϕ}} (d) {ϕ, {ϕ}, {ϕ, {ϕ}}}
2 (a) The empty set has no elements, so its cardinality is 0. (b) This set has one element (the empty set), so its cardinality is 1. (c) This set has two elements, so its cardinality is 2. (d) This set has three elements, so its cardinality is (12%) Determine whether each of these sets is the power set of a set, where a and b are distinct elements. (a) ϕ (b) {ϕ, {a}} (c) {ϕ, {a}, {ϕ, a}} (d) {ϕ, {a}, {b}, {a, b}} (a) The power set of every set includes at least the empty set, so the power set cannot be empty. Thus ϕ is not the power set of any set. (b) This is the power set of {a}. (c) This set has three elements. Since 3 is not a power of 2, this set cannot be the power set of any set. (d) This is the power set of {a, b}. 4. (5%) Find the sets A and B if A B = {1, 5, 7, 8}, B A = {2, 10}, and A B = {3, 6, 9}. Since A = (A B) (A B), we conclude that A = {1, 5, 7, 8} {3, 6, 9} = {1, 3, 5, 6, 7, 8, 9}. Similarly B = (B A) (A B) = {2, 10} {3, 6, 9} = {2, 3, 6, 9, 10}. 5. (6%) Let A, B, and C be sets. Show that (A B) C = (A C) (B C). First suppose x is in the lefthand side. Then x must be in A but in neither B nor C. Thus x A C, but x / B C, so x is in the righthand side. Next suppose that x is in the righthand side. Thus x must be in A C and not in B C. The
3 first of these implies that x A and x / C. But now it must also be the case that x / B, since otherwise we would have x B C. Thus we have shown that x is in A but in neither B nor C, which implies that x is in the lefthand side. 6. (9%) Draw the Venn diagrams for each of these combinations of the sets A, B, and C. (a) A (B C) (b) A B C (c) (A B) (A C) (B C) (a) (b) (c) 7. (3%) Determine whether f is a function from Z to R if (a) f(n) = ±n
4 (b) f(n) = n (c) f(n) = 1/(n 2 4) (Please describe the reason.) (a) This is not a function because the rule is not welldefined. We do not know whether f(3) = 3 or f(3) = 3. For a function, it cannot be both at the same time. (b) This is a function. For all integers n, n is a welldefined real number. (c) This is not a function with domain Z, since for n = 2 (and also for n = 2) the value of f(n) is not defined by the given rule. In other words, f(2) and f( 2) are not specified since division by 0 makes no sense. 8. (4%) Determine whether each of these functions from Z to Z is onetoone. Please describe the reason. (a) f(n) = n 1 (b) f(n) = n (c) f(n) = n 3 (d) f(n) = n/2 (a) This is onetoone, since if n 1 1 = n 2 1, then n 1 = n 2. (b) This is not onetoone, since, for example, f(3) = f( 3) = 10. (c) This is onetoone, since if n 3 1 = n 3 2, then n 1 = n 2 (take the cube root of each side). (d) This is not onetoone, since, for example, f(3) = f(4) = (10%) Determine whether f : Z Z Z is onto if (a) f(m, n) = 2m n
5 (b) f(m, n) = m 2 n 2 (c) f(m, n) = m + n + 1 (d) f(m, n) = m n (e) f(m, n) = m 2 4 (Please describe the reason.) (a) This is clearly onto, since f(0, n) = n for every integer n. (b) This is not onto, since, for example, 2 is not in the range. To see this, if m 2 n 2 = (m n)(m + 2) = 2, then m and n must have the same parity (both even or both odd). In either case, both m n and m + n are then even, so this expression is divisible by 4 and hence cannot equal 2. (c) This is onto, since f(0, n 1) = n for every integer n. (d) This is onto. To achieve negative values we set m = 0, and to achieve nonnegative values we set n = 0. (e) This is not onto, for the same reason as in part (b). 10. (8%) Determine whether each of these functions is a bijection from R to R. (a) f(x) = 3x + 4 (b) f(x) = 3x (c) f(x) = (x + 1)/(x + 2) (d) f(x) = x (a) This is a bijection since the inverse function is f 1 (x) = (4 x)/3. (b) This is not onetoone since f(17) = f( 17), for instance. It is also not onto, since the range is the interval (, 7]. For example, is not in the range.
6 (c) This function is a bijection, but not from R to R. To see that the domain and range are not R, note that x = 2 is not in the domain, and f(x) = 1 is not in the range. On the other hand, f is a bijection from R { 2} to R {1}, since its inverse is f 1 (x) = (1 2x)/(x 1) (d) It is clear that this continuous function is increasing throughout its entire domain (R) and it takes on both arbitrarily large values and arbitrarily small (large negative) ones. So it is a bijection. Its inverse is clearly f 1 (x) = 5 x (4%) Find f g and g f, where f(x) = x and g(x) = x + 2, are functions from R to R. We have (f g)(x) = f(g(x)) = f(x + 2) = (x + 2) = x 2 + 4x + 5, whereas (g f)(x) = g(f(x)) = g(x 2 + 1) = x = x Note that they are not eual. 12. (8%) For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. (a) 3, 6, 11, 18, 27, 38, 51, 66, 83, 102,... (b) 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011,... (c) 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682,... (d) 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1,... (a) The nth term is n The next three terms are 123, 146, 171. (b) The nth term is the binary expansion of n. The next three terms are 1100, 1101, (c) The nth term is 3 n 1, the next three terms are 59048, , (d) The sequence consists of one 1, followed by two 0s, then three 1s, four 0s, five 1s, and so on, alternating between 0s and 1s and having one more item in each group than in the previous group. Thus six 0 s will follow next, so the next three terms aer 0,0,0.
7 13. (4%) Compute each of these double sums. (a) (b) (c) (d) 3 2 (i j) i=1 j=1 3 2 (3i + 2j) i=0 j=0 3 2 j i=1 j=0 2 3 i 2 j 3 i=0 j=0 (a) 3 (b) 78 (c) 9 (d) (7%) Show that the polynomial function Z + Z + Z + with f(m, n) = (m + n 2)(m + n 1)/2 + m is onetoone and onto. f(1, 1) = 1 f(2, 1) = 3 f(3, 1) = 6 f(4, 1) = 10 f(5, 1) = 15 f(6, 1) = 21 f(1, 2) = 2 f(2, 2) = 5 f(3, 2) = 9 f(4, 2) = 14 f(5, 2) = 20 f(6, 2) = 27 f(1, 3) = 4 f(2, 3) = 8 f(3, 3) = 13 f(4, 3) = 19 f(5, 3) = 26 f(1, 4) = 7 f(2, 4) = 12 f(3, 4) = 18 f(4, 4) = 25 f(1, 5) = 11 f(2, 5) = 17 f(3, 5) = 24 f(1, 6) = 16 f(2, 6) = 23 f(1, 7) = 22 We see by looking at the diagonals of this table that the function takes on successive values as m + n increases. When m + n = 2, f(m, n) = 1. When m + n = 3, f(m, n) takes on the values 2 and 3. When m + n = 4, f(m, n) takes on the values
8 4, 5, and 6. And so on. It is clear from the formula that the range of values the function takes on for a fixed value of m + n, say m + n = x, is (x 2)(x 1) through (x 2)(x 1) 2 + (x 1), since m can assume the values 1,2,3,..., (x 1) under these conditions, and the first term in the formula is a fixed positive integer when m + n is fixed. To show that this function is onetoone and onto, we merely need to show that the range of values for x + 1 picks up precisely where the range of values for x left off, i.e., that f(x 1, 1) + 1 = f(1, x). We compute: f(x 1, 1)+1 = (x 2)(x 1) 2 + (x 1) +1 = x2 x = (x 1)x 2 +1 = f(1, x)
9 Extra Exercises: 1. Determine whether these statements are true or false. (a) ϕ {ϕ} (b) ϕ {ϕ, {ϕ}} (c) {ϕ} {ϕ} (d) {ϕ} {{ϕ}} (e) {ϕ} {ϕ, {ϕ}} (f) {{ϕ}} {ϕ, {ϕ}} (g) {{ϕ}} {{ϕ}, {ϕ}} (a) true. (b) true. (c) false see part (a). (d) true. (e) true the one element in the set on the left is an element of the set on the right, and the sets are not equal. (f) true similar to part (e). (g) false the two sets are equal. 2. Let A = {a, b, c}, B = {x, y}, C = {0, 1}. Find (a) A B C (b) C B A (c) C A B (d) B B B
10 (a) {(a, x, 0), (a, x, 1), (a, y, 0), (a, y, 1), (b, x, 0), (b, x, 1), (b, y, 0), (b, y, 1), (c, x, 0), (c, x, 1), (c, y, 0), (c, y, 1)} (b) {(0, x, a), (0, x, b), (0, x, c), (0, y, a), (0, y, b), (0, y, c), (1, x, a), (1, x, b), (1, x, c), (1, y, a), (1, y, b), (1, y, c)} (c) {(0, a, x), (0, a, y), (0, b, x), (0, b, y), (0, c, x), (0, c, y), (1, a, x), (1, a, y), (1, b, x), (1, b, y), (1, c, x), (1, c, y)} (d) {(x, x, x), (x, x, y), (x, y, x), (x, y, y), (y, x, x), (y, x, y), (y, y, x), (y, y, y)} 3. Prove the first De Morgan law by showing that if A and B are sets, A B = A B. A B = {x x / A B} = {x (x (A B))} = {x (x A x B)} = {x (x A) (x B)} = {x x / A x / B} = {x x A x B} = {x x A B} = A B 4. Show that the set Z + Z + is countable. Problem 14 gave us a onetoone correspondence between Z + Z + and Z +. Since Z + is countable, so is Z + Z +.
Math/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta
Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 4167362100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019 1
More informationEECS 203: Discrete Mathematics Individual Homework 3 (Winter 2016)
EECS 203: Discrete Mathematics Individual Homework 3 (Winter 2016) Answer Key Due: May 19, 2016 1. (2 points) Section 1.6 Problem 28 1. x(p (x) Q(x)) premise 2. x(( P (x) Q(x)) R(x)) premise Let c be an
More informationSequences and Summations. Niloufar Shafiei
Sequences and Summations Niloufar Shafiei Sequences A sequence is a discrete structure used to represent an ordered list. 1 Sequences A sequence is a function from a subset of the set integers (usually
More informationDiscrete Mathematics. Sec
Islamic University of Gaza Faculty of Engineering Department of Computer Engineering Fall 2011 ECOM 2311: Discrete Mathematics Eng. Ahmed Abumarasa Discrete Mathematics Sec 2.12.4 Basic Structures: Sets,
More information1 The Concept of a Mapping
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing
More informationIntroducing Functions
Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20
CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 23
CS 70 Discrete Mathematics for CS Spring 008 David Wagner Note 3 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of f ) to elements of set
More informationCardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.
Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection
More informationLecture 5. Introduction to Set Theory and the Pigeonhole Principle
Lecture 5. Introduction to Set Theory and the Pigeonhole Principle A set is an arbitrary collection (group) of the objects (usually similar in nature). These objects are called the elements or the members
More informationCS 441 Discrete Mathematics for CS Lecture 9. Functions II. CS 441 Discrete mathematics for CS. Functions
CS 441 Discrete Mathematics for CS Lecture 9 Functions II Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Functions Definition: Let and B be two sets. function from to B, denoted f : B, is an assignment
More informationCountable and Uncountable Sets
Countable and Uncountable Sets What follows is a different, and I hope simpler, presentation of the material in Appendix 3 of Grimaldi. For practice problems you should do Exercises A3, page A34, 16,
More informationChapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.
Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to
More informationModule MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions
Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective
More informationMATH 2000 ASSIGNMENT 9 SOLUTIONS
MATH 2000 ASSIGNMENT 9 SOLUTIONS 1. Let f : A B be a function. Write definitions for the following in logical form, with negations worked through. (a) f is onetoone iff x, y A, if f(x) = f(y) then x
More informationEquivalence Relations
Equivalence Relations A relation that is reflexive, transitive, and symmetric is called an equivalence relation. For example, the set {(a, a), (b, b), (c, c)} is an equivalence relation on {a, b, c}. An
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationChapter 4 FUNCTIONS. dom f = {a A : b B (a, b) f } ran f = {b B : a A (a, b) f }
Chapter 4 FUNCTIONS Until now we have learnt how to prove statements, we have introduced the basics of set theory and used those concepts to present and characterize the main features of the sets of interest
More informationRosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples
Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 2.4 Sequences and Summations Page references correspond to locations of Extra Examples icons in the textbook. p.52,
More informationSection Summary. Definition of sets Describing Sets
Chapter 2 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
More informationChapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since
Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the
More information2. Prove that the set of rational numbers with denominator 2 is countable. 3. Prove that the set of rational numbers with denominator 3 is countable.
Countable Sets Definition. A set X is called countable if X N. Exercise: 1. Prove that the set of even integers is 2. Prove that the set of rational numbers with denominator 2 is 3. Prove that the set
More informationSets and functions. {x R : x > 0}.
Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.
More informationLOGIC & SET THEORY AMIN WITNO
LOGIC & SET THEORY AMIN WITNO.. w w w. w i t n o. c o m Logic & Set Theory Revision Notes and Problems Amin Witno Preface These notes are for students of Math 251 as a revision workbook
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationFoundations of Computing Discrete Mathematics Solutions to exercises for week 2
Foundations of Computing Discrete Mathematics Solutions to exercises for week 2 Agata Murawska (agmu@itu.dk) September 16, 2013 Note. The solutions presented here are usually one of many possiblities.
More information3.2 Inverse Functions and Logarithmic Functions
Arkansas Tech University MATH 294: Calculus I Dr. Marcel B. Finan 3.2 Inverse Functions and Logarithmic Functions In this section we develop the derivative of an invertible function. As an application,
More information0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2.
SOLUTION SET FOR THE HOMEWORK PROBLEMS Page 5. Problem 8. Prove that if x and y are real numbers, then xy x + y. Proof. First we prove that if x is a real number, then x 0. The product of two positive
More informationArkansas Council of Teachers of Mathematics 2014 State Exam Algebra II
Arkansas Council of Teachers of Mathematics 2014 State Exam Algebra II For questions 1 through 25, mark your answer choice on the answer sheet provided. After completing items 1 through 25, answer each
More informationSets and Functions. Table of contents
Sets and Functions Table of contents 1. Sets..................................................... 2 1.1. Relations between sets...................................... 3 1.2. Operations that create new sets................................
More informationCountable and Uncountable Sets
Countable and Uncountable Sets In this section we extend the idea of the size of a set to infinite sets. It may come as somewhat of a surprise that there are different sizes of infinite sets. At the end
More informationIn mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)
Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics
More informationFinite and Infinite Sets
Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following
More informationFUNCTIONS. 1. Fundamentals
FUNCTIONS F 1. Fundamentals Suppose that we have two nonempty sets X and Y (keep in mind that these two sets may be equal). A function that maps X to Y is rule that associates to each element x X one
More information2.2. Functions. Figure 2.8. Correspondence x ± x. This operation is called a correspondence.
.. FUNCTIONS 7.. Functions... Correspondences. Suppose that to each element of a set we assign some elements of another set. For instance, = N, = Z, and to each element x N we assign all elements y Z such
More informationExample. (A sequence defined by induction or defined recursively ) a 1 = 1 = 1/1 and if a n = b n /c n where b n, c n Z, then
Chapter 2: (Infinite) Sequences and Series A(n infinite) sequence is a list of numbers a 1, a 2, a 3,..., indexed by N (or a 0, a 1, a 2, a 3,..., indexed by the nonnegative integers N 0 ) so there is
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationTHE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION. In Memory of Robert Barrington Leigh. Saturday, March 5, 2016
THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh Saturday, March 5, 2016 Time: 3 1 2 hours No aids or calculators permitted. The grading is designed
More information5 The Beginning of Transcendental Numbers
5 The Beginning of Transcendental Numbers We have defined a transcendental number (see Definition 3 of the Introduction), but so far we have only established that certain numbers are irrational. We now
More informationAxiom of Extensionality Two sets A and B are equal if and only if they have the same elements, i.e. A B and B A.
1. Sets, Relations, and Functions We begin by recalling our axioms, definitions, and concepts from Set Theory. We consider the first order logic whose objects are sets and which has two binary predicates:
More informationFunctions. CSE 215, Foundations of Computer Science Stony Brook University
Functions CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 A function f from a set X to a set Y f : X Y X is the domain Y is the codomain 1. every element
More informationDiscrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 39
CS 70 Discrete Mathematics for CS Fall 006 Papadimitriou & Vazirani Lecture 39 Infinity and Countability Consider a function f that maps elements of a set A (called the domain of f ) to elements of set
More informationSETS. Chapter Overview
Chapter 1 SETS 1.1 Overview This chapter deals with the concept of a set, operations on sets.concept of sets will be useful in studying the relations and functions. 1.1.1 Set and their representations
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationMTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions
MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions OUTLINE (References: Iyer Textbook  pages 4,6,17,18,40,79,80,81,82,103,104,109,110) 1. Definition of Function and Function Notation 2. Domain and Range
More informationDiscrete Mathematics Lecture 2 Logic of Quantified Statements, Methods of Proof, Set Theory, Number Theory Introduction and General Good Times
Discrete Mathematics Lecture 2 Logic of Quantified Statements, Methods of Proof, Set Theory, Number Theory Introduction and General Good Times Harper Langston New York University Predicates A predicate
More informationIf f is a 11 correspondence between A and B then it has an inverse, and f 1 isa 11 correspondence between B and A.
Chapter 5 Cardinality of sets 51 11 Correspondences A 11 correspondence between sets A and B is another name for a function f : A B that is 11 and onto If f is a 11 correspondence between A and B,
More informationMath 320 Course Notes. Chapter 7: Countable and Uncountable Sets
Math 320 Course Notes Magnhild Lien Chapter 7: Countable and Uncountable Sets Hilbert s Motel: Imagine a motel with in nitely many rooms numbered 1; 2; 3; 4 ; n; : One evening an "in nite" bus full with
More informationMathematics for Computer Science
Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math
More information1 SET THEORY CHAPTER 1.1 SETS
CHAPTER 1 SET THEORY 1.1 SETS The main object of this book is to introduce the basic algebraic systems (mathematical systems) groups, ring, integral domains, fields, and vector spaces. By an algebraic
More informationEquivalence relations
Equivalence relations A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a fraction a/b, a, b Z and b 0, and hence
More informationThe composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result.
30 5.6 The chain rule The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result." Example. g(x) = x 2 and f(x) = (3x+1).
More informationProposition 11 Let A be a set. The following are equivalent.
Countable Sets Definition 6 Let A be a set It is countably infinite if N A and is countable if it is finite or countably infinite If it is not countable it is uncountable Note that if A B then A is countable
More informationMath 115 Spring 2014 Written Homework 8SOLUTIONS Due Friday, April 11
Math 115 Spring 2014 Written Homework 8SOLUTIONS Due Friday, April 11 Instructions: Write complete solutions on separate paper (not spiral bound). If multiple pieces of paper are used, THEY MUST BE STAPLED
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationSETS, RELATIONS, AND FUNCTIONS
September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four
More informationRadical Expressions and Graphs
Radical Expressions and Graphs Objective 1 Find square roots. Slide 10.13 Find square roots. When squaring a number, multiply the number by itself. To find the square root of a number, find a number that
More information15251: Great Theoretical Ideas in Computer Science Lecture 6. To Infinity and Beyond
15251: Great Theoretical Ideas in Computer Science Lecture 6 To Infinity and Beyond Galileo (1564 1642) Best known publication: Dialogue Concerning the Two Chief World Systems His final magnum opus (1638):
More information1. A permutation of the set A is a bijection from A to itself in other words a function α : A A such that α is a bijection (onetoone and onto).
Chapter 6 Permutation Groups 6.1 Definitions and Array Notation In this chapter, we will study transformations which reshuffl e the elements of a set. Mathematically, these transformations are bijections
More informationChapter 17. Countability The rationals and the reals. This chapter covers infinite sets and countability.
Chapter 17 Countability This chapter covers infinite sets and countability. 17.1 The rationals and the reals You re familiar with three basic sets of numbers: the integers, the rationals, and the reals.
More informationCountable and Uncountable Sets
Countable and Uncountable Sets Rich Schwartz November 12, 2007 The purpose of this handout is to explain the notions of countable and uncountable sets. 1 Basic Definitions A map f between sets S 1 and
More informationThis chapter describes set theory, a mathematical theory that underlies all of modern mathematics.
Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.
More informationWe give a basic overview of the mathematical background required for this course.
1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The
More informationDefinition 1 (Informal definition). The limit of a function f(x), as x approaches a R, is L R, and we write. lim
Formal vs Informal Definition of a Limit Definition (Informal definition). The it of a function f(x), as x approaches a R, is L R, and we write L of x close enough to a. Definition 2 (Formal definition).
More informationSets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use
More informationMATH / Assignment 2. November 6, 2002 Late penalty: 5% for each school day.
MATH 260 2002/20013 Assignment 2 November 6, 2002 Late penalty: 5% for each school day. 1. 1.6 #. (2 points each part) Find the domain and range of the following functions. (a) the function that assigns
More informationIt is not immediately obvious that this should even give an integer. Since 1 < 1 5
Math 163  Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationSection 1.1 Real numbers. Set Builder notation. Interval notation
Section 1.1 Real numbers Set Builder notation Interval notation Functions a function is the set of all possible points y that are mapped to a single point x. If when x=5 y=4,5 then it is not a function
More informationTopology of the Real Numbers
Chapter 5 Topology of the Real Numbers In this chapter, we define some topological properties of the real numbers R and its subsets. 5.1. Open sets Open sets are among the most important subsets of R.
More informationMathematics Department Stanford University Math 61CM Homework 5
Mathematics Department Stanford University Math 61CM Homework 5 Solutions 1. Suppose A is an m n real matrix. Prove: (a) A T A is positive semidefinite, and (b) A T A is positive definite if N(A) {0}.
More information1 Partitions and Equivalence Relations
Today we re going to talk about partitions of sets, equivalence relations and how they are equivalent. Then we are going to talk about the size of a set and will see our first example of a diagonalisation
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More informationDefinition. A function f from A to B is a relation from A to B such that:
Functions Definition A function f from A to B is a relation from A to B such that: (i) Dom(f) = A, and (ii) If (x,y) f and (x,z) f then y = z. If A = B, we say that f is a function on A. In terms of ordered
More informationS(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.
MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ  http://www.inf.unibz.it/ artale/ SECTION 5.2 Mathematical Induction I Copyright Cengage Learning. All rights reserved.
More information2.8 Cardinality Introduction Finite and Infinite Sets 2.8. CARDINALITY 73
2.8. CARDINALITY 73 2.8 Cardinality 2.8.1 Introduction Cardinality when used with a set refers to the number of elements the set has. In this section, we will learn how to distinguish between finite and
More informationChapter 1 Summary. x2 +x y 1
Chapter Summary. Distance Formula The distance between two points P = (x,y ) and P = (x,y ) in the xyplane is Midpoint Formula P P = (x x ) +(y y ) The midpoint between two points P = (x,y ) and P = (x,y
More informationAnnouncements. CompSci 230 Discrete Math for Computer Science. Test 1
CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 47 Prof. Rodger is out Sep 30Oct 4 There is Recitation: Sept 2730.
More informationIntroduction to Series and Sequences Math 121 Calculus II D Joyce, Spring 2013
Introduction to Series and Sequences Math Calculus II D Joyce, Spring 03 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial
More informationSets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.
Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in
More information1.5 Function Notation
44 Relations and Functions 1.5 Function Notation In Definition 1.4, we described a function as a special kind of relation one in which each x coordinate is matched with only one ycoordinate. In this
More informationMath 430 Problem Set 1 Solutions
Math 430 Problem Set 1 Solutions Due January 22, 2016 1.2. If A = {a, b, c}, B = {1, 2, 3}, C = {x}, and D =, list all of the elements of each of the following sets. (a) A B (b) B A {(a, 1), (a, 2), (a,
More informationLECTURE NOTES ON RELATIONS AND FUNCTIONS
LECTURE NOTES ON RELATIONS AND FUNCTIONS PETE L. CLARK Contents 1. Relations 1 1.1. The idea of a relation 1 1.2. The formal definition of a relation 2 1.3. Basic terminology and further examples 2 1.4.
More informationFunctions between Sets
Chapter 3 Functions between Sets 3.1 Functions 3.1.1 Functions, Domains, and Codomains In the previous chapter, we investigated the basics of sets and operations on sets. In this chapter, we will analyze
More informationf n = n is a oneto one correspondence from N
Section.5 1 Section.5 Purpose of Section To introduce the concept of uncountable sets. We present Cantor s proof that the real numbers are uncountable. We also show that the cardinality of the plane is
More informationALGEBRA QUALIFYING EXAM PROBLEMS RING THEORY
ALGEBRA QUALIFYING EXAM PROBLEMS RING THEORY Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2012 CONTENTS RING THEORY General
More informationMATH S104 Lecture 6 Inclass Problem Solutions
MATH S104 Lecture 6 Inclass Problem Solutions July 9, 2009 Problem 1 Prove that the recursive factorial algorithm is correct. Algorithm 4.0: A recursive algorithm for computing n! procedure f actorial(n
More informationMath Chapter 1 Essentials of Calculus by James Stewart Prepared by Jason Gaddis
Math 23  Chapter Essentials of Calculus by James Stewart Prepared by Jason Gaddis Chapter  Functions.  Functions and their representations Definition A function is a rule that assignes to each element
More informationPUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.
PUTNAM TRAINING NUMBER THEORY (Last updated: November 8, 2016) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice
More informationREAL ANALYSIS I HOMEWORK 2
REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other
More informationHonors Math 2 Final Exam
Name Teacher Block Honors Math 2 Final Exam Lexington High School June 18, 2014 General directions: Show complete work supporting your answers. Please circle final answers where appropriate. Decimal values
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationMath 31A  Review of Precalculus
Math 31A  Review of Precalculus 1 Introduction One frequent complaint voiced by reluctant students of mathematics is: Why do I have to study this? When will I need math in real life? While it is my personal
More informationReview of Key Concepts: 1.2 Characteristics of Polynomial Functions
Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Polynomial functions of the same degree have similar characteristics The degree and leading coefficient of the equation of the polynomial
More informationCHAPTER 2. Inequalities
CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential
More informationSolutions to Practice Final 1
s to Practice Final 1 1. (a) What is φ(0 100 ) where φ is Euler s φfunction? (b) Find an integer x such that 140x 133 (mod 301). Hint: gcd(140, 301) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100
More informationLecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties
Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an
More information