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

Size: px
Start display at page:

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

Transcription

1 Chapter 4

2 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 For every element a A there is b B such that (a, b) f. If (a, b),(a, c) f then b = c.

3 We write f : A B if f is a function from A to B. If f : A B thena is called the domain of f and B is called the co-domain of f. The image of a set S A by a function f : A B is the set f(s) = {f(a) a S}. The pre-image of a set T B by a function f : A B is the set f 1 (T) = {a f(a) T }.

4 Some Examples Polynomial function f : R R given by f(x) = n i=1 a i x i where a i R and n N {0}. f : R R given by f(x) = x if x is rational and f(x) = 0 if x is irrational. f : R Z given by f(x) is the least integer which is greater than or equal to x. (Ceiling function). f : R Z given by f(x) is the largest integer which is smaller than or equal to x. (Floor function).

5 Characteristic function. Let A U, f : U {0,1} given by f(x) = 1 if x A and f(x) = 0 if x / A. f : A P(A) given by f(x) = A {x}. f : P(Z + ) Z + {0} given by f(a) is the smallest integer z with z A if Z and f( ) = 0. f : Z + Z + given by f(x) = f(f(x/2)) if x is even and 1 if x is odd.

6 When are functions f and g equal? Functions f, g are equal if f : A B, g : A B and for every a A, f(a) = g(a), that is the sets f, g A B are such that f = g.

7 Injective, surjective, bijective functions Definition 3 Let f : A B. f is called injective (one-to-one) if for every x, y A if f(x) = f(y) then x = y. f is called surjective (onto) if for every b B there is a A with f(a) = b. f is called a bijection (one-to-one correspondence) if f is both injective and surjective.

8 Injection Surjection Bijection

9 Example 1 Check which of the functions below are injective, which are surjective. f : N N, f(x) = x + 7. f : Z Z, f(x) = x + 7. f : N N, f(x) = x f : Z Z, f(x) = x f : N N, f(x) = x 5.

10 f : N N Z, f((x, y)) = x y. f : Z Z Z, f((x, y)) = x + y + 3. f : Z Z Z, f(x) = (x,3x). f : R R, f(x) = x 2 1 if x 0 and f(x) = x 2 if x < 0. f : R R, f(x) = x 2 if x 1 and f(x) = 1 + x if x < 1.

11 Increasing and decreasing functions Definition 4 Let A, B R and let f : A B. f is called increasing when for every x, y A, if x < y then f(x) < f(y). f is called decreasing when for every x, y A, if x < y then f(x) > f(y).

12 Composition of functions Definition 5 Let f : A B and g : B C. Then g f is a function from A to C defined as follows (g f)(x) = g(f(x)). Example 2 Show that f : A B and g : B C are both surjective then g f : A C is surjective. Show that f : A B and g : B C are both injective then g f : A C is injective.

13 Example 3 Decide if the statements are true or false. Let f : A B and g : B C. If g f is injective then f is injective. If g f is injective then g is injective. If g f is surjective then f is surjective. If g f is surjective then g is surjective.

14 Inverse of a function We say that a function f : B is invertible if there exists a function g : B A such that for all a A, b B f(a) = b if and only if g(b) = a. Theorem 1 Let f : A B. Then f is invertible if and only if it is a bijection. Moreover, if f has an inverse then it is unique. If f : A B then the inverse of f is denoted by f 1. Note that f f 1 = id B, f 1 f = id A.

15 Proof of Theorem 1. Suppose g is an inverse of f. Show that f must be injective and surjective. Suppose f is a bijection. Define g(b) to be equal to a such that f(a) = b. Check that g is a function. Suppose g 1, g 2 are inverses of f. Show that g 1 = g 2.

16 Denumerable sets When two finite sets A, B have same cardinality? When there is a bijection f : A B. Definition 6 Two sets A, B are equipollent if there is a bijection f : A B. Note: If A is equipollent to B then B is equipollent to A. A set A is called denumerable if it is equipollent with N. A set A is called countable if it is either finite or denumerable.

17 N is denumerable. Z is denumerable. The set of even integers is denumerable. The set {5 + 1/n n N} is denumerable.

18 (a) Any infinite subset of N is denumerable. (b) Any infinite subset of a denumerable set is denumerable.

19 (a) N N is denumerable. Use f : N N N, f((a, b)) = 2 a 1 (2b 1). (b) Q + is denumerable. Use f : Q + N N, f(p/q) = (p, q) assuming p, q have no common factors.

20 A set A is called uncountable if it is not countable. Theorem 2 [Cantor s Theorem] The set of real numbers is uncountable. Georg Cantor, ,

21 Proof. Can you enumerate all possible infinite strings of blue/red marbles?

22 Theorem 3 For every set X, X and P(X) are not equipollent.

23 Proof. Show there is no surjection from X to P(X). By contradiction suppose f : X P(X) is a surjection. Let Y = {x X x / f(x)}. Let y X be such that f(y) = Y. If y f(y) = Y then by definition of Y, y / Y. If y / f(y) = Y then be definition of Y, y Y.

1.4 Cardinality. Tom Lewis. Fall Term Tom Lewis () 1.4 Cardinality Fall Term / 9

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

More information

2.1 Sets. Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R.

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

More information

CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH. Fall CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Define and compute the cardinality of a set. Use functions to compare the sizes of sets. Classify sets

More information

MATH 3300 Test 1. Name: Student Id:

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

More information

MATH 201 Solutions: TEST 3-A (in class)

MATH 201 Solutions: TEST 3-A (in class) MATH 201 Solutions: TEST 3-A (in class) (revised) God created infinity, and man, unable to understand infinity, had to invent finite sets. - Gian Carlo Rota Part I [5 pts each] 1. Let X be a set. Define

More information

SETS AND FUNCTIONS JOSHUA BALLEW

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,

More information

A Short Review of Cardinality

A Short Review of Cardinality Christopher Heil A Short Review of Cardinality November 14, 2017 c 2017 Christopher Heil Chapter 1 Cardinality We will give a short review of the definition of cardinality and prove some facts about the

More information

Discussion Summary 10/16/2018

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

More information

A Readable Introduction to Real Mathematics

A Readable Introduction to Real Mathematics Solutions to selected problems in the book A Readable Introduction to Real Mathematics D. Rosenthal, D. Rosenthal, P. Rosenthal Chapter 10: Sizes of Infinite Sets 1. Show that the set of all polynomials

More information

Definition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.

Definition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS. 4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be

More information

Chapter 1 : The language of mathematics.

Chapter 1 : The language of mathematics. MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :

More information

Mathematics 220 Workshop Cardinality. Some harder problems on cardinality.

Mathematics 220 Workshop Cardinality. Some harder problems on cardinality. Some harder problems on cardinality. These are two series of problems with specific goals: the first goal is to prove that the cardinality of the set of irrational numbers is continuum, and the second

More information

MATH 13 FINAL EXAM SOLUTIONS

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

More information

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 27

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 27 CS 70 Discrete Mathematics for CS Spring 007 Luca Trevisan Lecture 7 Infinity and Countability Consider a function f that maps elements of a set A (called the domain of f ) to elements of set B (called

More information

Discrete Mathematics 2007: Lecture 5 Infinite sets

Discrete Mathematics 2007: Lecture 5 Infinite sets Discrete Mathematics 2007: Lecture 5 Infinite sets Debrup Chakraborty 1 Countability The natural numbers originally arose from counting elements in sets. There are two very different possible sizes for

More information

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 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

More information

Undecidability. Andreas Klappenecker. [based on slides by Prof. Welch]

Undecidability. Andreas Klappenecker. [based on slides by Prof. Welch] Undecidability Andreas Klappenecker [based on slides by Prof. Welch] 1 Sources Theory of Computing, A Gentle Introduction, by E. Kinber and C. Smith, Prentice-Hall, 2001 Automata Theory, Languages and

More information

Final Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is

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,

More information

Week Some Warm-up Questions

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

More information

CITS2211 Discrete Structures (2017) Cardinality and Countability

CITS2211 Discrete Structures (2017) Cardinality and Countability CITS2211 Discrete Structures (2017) Cardinality and Countability Highlights What is cardinality? Is it the same as size? Types of cardinality and infinite sets Reading Sections 45 and 81 84 of Mathematics

More information

Cardinality of sets. Cardinality of sets

Cardinality of sets. Cardinality of sets Cardinality of sets Two sets A and B have the same size, or cardinality, if and only if there is a bijection f : A Ñ B. Example: We know that set ta, b, cu has elements because we can count them: 1: a

More information

Math.3336: Discrete Mathematics. Cardinality of Sets

Math.3336: Discrete Mathematics. Cardinality of Sets Math.3336: Discrete Mathematics Cardinality of Sets Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall 2018

More information

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. 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.

More information

LECTURE 22: COUNTABLE AND UNCOUNTABLE SETS

LECTURE 22: COUNTABLE AND UNCOUNTABLE SETS LECTURE 22: COUNTABLE AND UNCOUNTABLE SETS 1. Introduction To end the course we will investigate various notions of size associated to subsets of R. The simplest example is that of cardinality - a very

More information

Solutions to Homework Problems

Solutions to Homework Problems Solutions to Homework Problems November 11, 2017 1 Problems II: Sets and Functions (Page 117-118) 11. Give a proof or a counterexample of the following statements: (vi) x R, y R, xy 0; (x) ( x R, y R,

More information

Functions as Relations

Functions as Relations Functions as Relations Definition Recall that if A and B are sets, then a relation from A to B is a subset of A B. A function from A to B is a relation f from A to B with the following properties (i) The

More information

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.

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

More information

1 Partitions and Equivalence Relations

1 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 information

S15 MA 274: Exam 3 Study Questions

S15 MA 274: Exam 3 Study Questions S15 MA 274: Exam 3 Study Questions You can find solutions to some of these problems on the next page. These questions only pertain to material covered since Exam 2. The final exam is cumulative, so you

More information

Functions. Given a function f: A B:

Functions. Given a function f: A B: Functions Given a function f: A B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f.

More information

Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 2 Solutions Please write neatly, and in complete sentences when possible.

Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 2 Solutions Please write neatly, and in complete sentences when possible. Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 2 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 1.4.2, 1.4.4, 1.4.9, 1.4.11,

More information

Section 4.4 Functions. CS 130 Discrete Structures

Section 4.4 Functions. CS 130 Discrete Structures Section 4.4 Functions CS 130 Discrete Structures Function Definitions Let S and T be sets. A function f from S to T, f: S T, is a subset of S x T where each member of S appears exactly once as the first

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2014 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

9/21/2018. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions

9/21/2018. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions How can we prove that a function f is one-to-one? Whenever you want to prove something, first take a look at the relevant definition(s): x, y A (f(x) = f(y) x = y) f:r R f(x) = x 2 Disproof by counterexample:

More information

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements

More information

Math 105A HW 1 Solutions

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

More information

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 MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A

More information

One-to-one functions and onto functions

One-to-one functions and onto functions MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are

More information

MATH FINAL EXAM REVIEW HINTS

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

More information

Section Summary. Definition of a Function.

Section Summary. Definition of a Function. Section 2.3 Section Summary Definition of a Function. Domain, Cdomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial

More information

Cardinality and ordinal numbers

Cardinality and ordinal numbers Cardinality and ordinal numbers The cardinality A of a finite set A is simply the number of elements in it. When it comes to infinite sets, we no longer can speak of the number of elements in such a set.

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE B.Sc. MATHEMATICS V SEMESTER (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS QUESTION BANK 1. Find the number of elements in the power

More information

Notes. Functions. Introduction. Notes. Notes. Definition Function. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y.

Notes. Functions. Introduction. Notes. Notes. Definition Function. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Functions Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Section 2.3 of Rosen cse235@cse.unl.edu Introduction

More information

9/21/17. Functions. CS 220: Discrete Structures and their Applications. Functions. Chapter 5 in zybooks. definition. definition

9/21/17. Functions. CS 220: Discrete Structures and their Applications. Functions. Chapter 5 in zybooks. definition. definition Functions CS 220: Discrete Structures and their Applications Functions Chapter 5 in zybooks A function maps elements from one set X to elements of another set Y. A B Brian C Drew range: {A, B, D} Alan

More information

MATH 13 SAMPLE FINAL EXAM SOLUTIONS

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.

More information

Functions Functions and Modeling A UTeach/TNT Course

Functions Functions and Modeling A UTeach/TNT Course Definition of a Function DEFINITION: Let A and B be sets. A function between A and B is a subset of A B with the property that if (a, b 1 )and(a, b 2 ) are both in the subset, then b 1 = b 2. The domain

More information

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 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

More information

CSE 20 Discrete Math. Winter, January 24 (Day 5) Number Theory. Instructor: Neil Rhodes. Proving Quantified Statements

CSE 20 Discrete Math. Winter, January 24 (Day 5) Number Theory. Instructor: Neil Rhodes. Proving Quantified Statements CSE 20 Discrete Math Proving Quantified Statements Prove universal statement: x D, P(x)Q(x) Exhaustive enumeration Generalizing from the generic particular Winter, 2006 Suppose x is in D and P(x) Therefore

More information

Selected problems from past exams

Selected problems from past exams Discrete Structures CS2800 Prelim 1 s Selected problems from past exams 1. True/false. For each of the following statements, indicate whether the statement is true or false. Give a one or two sentence

More information

Introduction to Decision Sciences Lecture 6

Introduction to Decision Sciences Lecture 6 Introduction to Decision Sciences Lecture 6 Andrew Nobel September 21, 2017 Functions Functions Given: Sets A and B, possibly different Definition: A function f : A B is a rule that assigns every element

More information

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k)

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k) MATH 111 Optional Exam 3 lutions 1. (0 pts) We define a relation on Z as follows: a b if a + b is divisible by 3. (a) (1 pts) Prove that is an equivalence relation. (b) (8 pts) Determine all equivalence

More information

Finite and Infinite Sets

Finite 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 information

Chapter 1. Sets and Mappings

Chapter 1. Sets and Mappings Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write

More information

MATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals

MATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals MATH 02 INTRODUCTION TO MATHEMATICAL ANALYSIS Properties of Real Numbers Some Fundamentals The whole course will be based entirely on the study of sequence of numbers and functions defined on the real

More information

Countable and uncountable sets. Matrices.

Countable and uncountable sets. Matrices. Lecture 11 Countable and uncountable sets. Matrices. Instructor: Kangil Kim (CSE) E-mail: kikim01@konkuk.ac.kr Tel. : 02-450-3493 Room : New Milenium Bldg. 1103 Lab : New Engineering Bldg. 1202 Next topic:

More information

General Notation. Exercises and Problems

General Notation. Exercises and Problems Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.

More information

Countable and uncountable sets. Matrices.

Countable and uncountable sets. Matrices. CS 441 Discrete Mathematics for CS Lecture 11 Countable and uncountable sets. Matrices. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Arithmetic series Definition: The sum of the terms of the

More information

CSE 311: Foundations of Computing. Lecture 26: Cardinality

CSE 311: Foundations of Computing. Lecture 26: Cardinality CSE 311: Foundations of Computing Lecture 26: Cardinality Cardinality and Computability Computers as we know them grew out of a desire to avoid bugs in mathematical reasoning A brief history of reasoning

More information

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. 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

More information

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 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

More information

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. 1. MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection

More information

Math 300: Final Exam Practice Solutions

Math 300: Final Exam Practice Solutions Math 300: Final Exam Practice Solutions 1 Let A be the set of all real numbers which are zeros of polynomials with integer coefficients: A := {α R there exists p(x) = a n x n + + a 1 x + a 0 with all a

More information

Func%ons. function f to the element a of A. Functions are sometimes called mappings or transformations.

Func%ons. function f to the element a of A. Functions are sometimes called mappings or transformations. Section 2.3 Func%ons Definition: Let A and B be nonempty sets. A function f from A to B, denoted f: A B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2016 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from the set A to the set B is a correspondence that assigns to

More information

University of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics

University of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics University of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics Examiners: D. Burbulla, P. Glynn-Adey, S. Homayouni Time: 7-10

More information

Discrete dynamics on the real line

Discrete dynamics on the real line Chapter 2 Discrete dynamics on the real line We consider the discrete time dynamical system x n+1 = f(x n ) for a continuous map f : R R. Definitions The forward orbit of x 0 is: O + (x 0 ) = {x 0, f(x

More information

4) Have you met any functions during our previous lectures in this course?

4) Have you met any functions during our previous lectures in this course? Definition: Let X and Y be sets. A function f from the set X to the set Y is a rule which associates to each element x X a unique element y Y. Notation: f : X Y f defined on X with values in Y. x y y =

More information

Great Theoretical Ideas in Computer Science. Lecture 5: Cantor s Legacy

Great Theoretical Ideas in Computer Science. Lecture 5: Cantor s Legacy 15-251 Great Theoretical Ideas in Computer Science Lecture 5: Cantor s Legacy September 15th, 2015 Poll Select the ones that apply to you: - I know what an uncountable set means. - I know Cantor s diagonalization

More information

Introduction to Proofs

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

More information

Functions and cardinality (solutions) sections A and F TA: Clive Newstead 6 th May 2014

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

More information

CS 220: Discrete Structures and their Applications. Functions Chapter 5 in zybooks

CS 220: Discrete Structures and their Applications. Functions Chapter 5 in zybooks CS 220: Discrete Structures and their Applications Functions Chapter 5 in zybooks Functions A function maps elements from one set X to elements of another set Y. A Brian Drew Alan Ben B C D F range: {A,

More information

Sec$on Summary. Definition of a Function.

Sec$on Summary. Definition of a Function. Section 2.3 Sec$on Summary Definition of a Function. Domain, Codomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) I Semester Core Course. FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) I Semester Core Course. FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) (2011 Admission Onwards) I Semester Core Course FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK 1) If A and B are two sets

More information

INFINITY: CARDINAL NUMBERS

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

More information

MAT115A-21 COMPLETE LECTURE NOTES

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

More information

ECS 120 Lesson 18 Decidable Problems, the Halting Problem

ECS 120 Lesson 18 Decidable Problems, the Halting Problem ECS 120 Lesson 18 Decidable Problems, the Halting Problem Oliver Kreylos Friday, May 11th, 2001 In the last lecture, we had a look at a problem that we claimed was not solvable by an algorithm the problem

More information

COM S 330 Homework 05 Solutions. Type your answers to the following questions and submit a PDF file to Blackboard. One page per problem.

COM S 330 Homework 05 Solutions. Type your answers to the following questions and submit a PDF file to Blackboard. One page per problem. Type your answers to the following questions and submit a PDF file to Blackboard. One page per problem. Problem 1. [5pts] Consider our definitions of Z, Q, R, and C. Recall that A B means A is a subset

More information

1 of 8 7/15/2009 3:43 PM Virtual Laboratories > 1. Foundations > 1 2 3 4 5 6 7 8 9 6. Cardinality Definitions and Preliminary Examples Suppose that S is a non-empty collection of sets. We define a relation

More information

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

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

More information

Fundamentals of Pure Mathematics - Problem Sheet

Fundamentals of Pure Mathematics - Problem Sheet Fundamentals of Pure Mathematics - Problem Sheet ( ) = Straightforward but illustrates a basic idea (*) = Harder Note: R, Z denote the real numbers, integers, etc. assumed to be real numbers. In questions

More information

NOTE: You have 2 hours, please plan your time. Problems are not ordered by difficulty.

NOTE: You have 2 hours, please plan your time. Problems are not ordered by difficulty. EXAM 2 solutions (COT3100, Sitharam, Spring 2017) NAME:last first: UF-ID Section NOTE: You have 2 hours, please plan your time. Problems are not ordered by difficulty. (1) Are the following functions one-to-one

More information

Intro to Theory of Computation

Intro to Theory of Computation Intro to Theory of Computation LECTURE 15 Last time Decidable languages Designing deciders Today Designing deciders Undecidable languages Diagonalization Sofya Raskhodnikova 3/1/2016 Sofya Raskhodnikova;

More information

447 HOMEWORK SET 1 IAN FRANCIS

447 HOMEWORK SET 1 IAN FRANCIS 7 HOMEWORK SET 1 IAN FRANCIS For each n N, let A n {(n 1)k : k N}. 1 (a) Determine the truth value of the statement: for all n N, A n N. Justify. This statement is false. Simply note that for 1 N, A 1

More information

Announcements. CS243: Discrete Structures. Sequences, Summations, and Cardinality of Infinite Sets. More on Midterm. Midterm.

Announcements. CS243: Discrete Structures. Sequences, Summations, and Cardinality of Infinite Sets. More on Midterm. Midterm. Announcements CS43: Discrete Structures Sequences, Summations, and Cardinality of Infinite Sets Işıl Dillig Homework is graded, scores on Blackboard Graded HW and sample solutions given at end of this

More information

Copyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction

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

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

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

More information

LESSON RELATIONS & FUNCTION THEORY

LESSON RELATIONS & FUNCTION THEORY 2 Definitions LESSON RELATIONS & FUNCTION THEORY Ordered Pair Ordered pair of elements taken from any two sets P and Q is a pair of elements written in small brackets and grouped together in a particular

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

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

More information

Sets are one of the basic building blocks for the types of objects considered in discrete mathematics.

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

More information

Section 7.5: Cardinality

Section 7.5: Cardinality Section 7: Cardinality In this section, we shall consider extend some of the ideas we have developed to infinite sets One interesting consequence of this discussion is that we shall see there are as many

More information

Math 127 Homework. Mary Radcliffe. Due 29 March Complete the following problems. Fully justify each response.

Math 127 Homework. Mary Radcliffe. Due 29 March Complete the following problems. Fully justify each response. Math 17 Homework Mary Radcliffe Due 9 March 018 Complete the following problems Fully justify each response NOTE: due to the Spring Break, this homework set is a bit longer than is typical You only need

More information

CSCE 222 Discrete Structures for Computing. Review for Exam 1. Dr. Hyunyoung Lee !!!

CSCE 222 Discrete Structures for Computing. Review for Exam 1. Dr. Hyunyoung Lee !!! CSCE 222 Discrete Structures for Computing Review for Exam 1 Dr. Hyunyoung Lee 1 Topics Propositional Logic (Sections 1.1, 1.2 and 1.3) Predicate Logic (Sections 1.4 and 1.5) Rules of Inferences and Proofs

More information

Section 1.7 Proof Methods and Strategy. Existence Proofs. xp(x). Constructive existence proof:

Section 1.7 Proof Methods and Strategy. Existence Proofs. xp(x). Constructive existence proof: Section 1.7 Proof Methods and Strategy Existence Proofs We wish to establish the truth of xp(x). Constructive existence proof: - Establish P(c) is true for some c in the universe. - Then xp(x) is true

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Spring

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Spring c Dr Oksana Shatalov, Spring 2016 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

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

More information

LECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel

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

More information

Date: October 24, 2008, Friday Time: 10:40-12:30. Math 123 Abstract Mathematics I Midterm Exam I Solutions TOTAL

Date: October 24, 2008, Friday Time: 10:40-12:30. Math 123 Abstract Mathematics I Midterm Exam I Solutions TOTAL Date: October 24, 2008, Friday Time: 10:40-12:30 Ali Sinan Sertöz Math 123 Abstract Mathematics I Midterm Exam I Solutions 1 2 3 4 5 TOTAL 20 20 20 20 20 100 Please do not write anything inside the above

More information

Discrete Structures for Computer Science

Discrete Structures for Computer Science Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #10: Sequences and Summations Based on materials developed by Dr. Adam Lee Today s Topics Sequences

More information

Section 0. Sets and Relations

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

More information

RED. Fall 2016 Student Submitted Sample Questions

RED. Fall 2016 Student Submitted Sample Questions RED Fall 2016 Student Submitted Sample Questions Name: Last Update: November 22, 2016 The questions are divided into three sections: True-false, Multiple Choice, and Written Answer. I will add questions

More information