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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 c Dr Oksana Shatalov, Fall 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 A one and only one element in the set B. We call A the domain of f and B the codomain of f. We write f : A B and for each a A we write f(a) = b if b is assigned to a. Using diagram EXAMPLE 2. Let A = {2, 4, 6, 10} and B = {0, 1, 1, 8}. Write out three functions with domain A and codomain B. Some common functions Identity function i A : A A maps every element to itself: Linear function f : R R is defined by Constant function f : R R is defined by

2 c Dr Oksana Shatalov, Fall DEFINITION 3. Two functions f and g are equal if they have the same domain and the same codomain and if f(a) = g(a) for all a in domain. Image of a Function EXAMPLE 4. Discuss codomain of f(x) = x 4. DEFINITION 5. Let f : A B be a function. The image of f is Im(f) = {y B y = f(x) for some x A}. Note that f(x) is the image of the set X under f. The graph of f is the set {(a, b) A B b = f(a)}. EXAMPLE 6. f : R R is defined by f(x) = cos x. Find Im(f) and f([0, π/2]). EXAMPLE 7. f : R R is defined by f(x) = cos x. Find Im(f). EXAMPLE 8. f : R Z is defined by f(x) = [x], where [x] means the greatest integer x. Find Im(f).

3 c Dr Oksana Shatalov, Fall Rules For f : A B to prove that Im(f) = S, use the following tautology: (y S) ( x A f(x) = y). For f : A B to prove that f(x) = S, X A, use the following tautology (y S) ( x X f(x) = y). EXAMPLE 9. Let S = {y R y 0}. Prove that if f : R R is defined by f(x) = x 4 then Im(f) = S.

4 c Dr Oksana Shatalov, Fall EXAMPLE 10. f : R R is defined by f(x) = 5x 4. Find f([0, 1]). Justify your answer. EXAMPLE 11. f : Z Z is defined by Prove that f(e) = O. f(n) = { n 1 if n is even, n + 1 if n is odd.

5 c Dr Oksana Shatalov, Fall PROPOSITION 12. Let A and B be nonempty sets and f : A B be a function. If X Y A then f(x) f(y ). Proof. PROPOSITION 13. Let A and B be nonempty sets and f : A B be a function. If X A and Y A then (a) f(x Y ) = f(x) f(y ). (b) f(x Y ) f(x) f(y ). Proof

6 c Dr Oksana Shatalov, Fall Inverse Image DEFINITION 14. Let f : A B be a function and let W be a subset of its codomain (i.e. W B). Then the inverse image of W (written f 1 (W )) is the set f 1 (W ) = {a A f(a) W }. REMARK 15. This definition implies the following: (x f 1 (W )) (f(x) W ) If W Im(f) then (S = f 1 (W )) (f(s) = W ) EXAMPLE 16. Let A = {a, b, c, d, e, f} and B = {7, 9, 11, 12, 13} and let the function g : A B be given by g(a) = 11, g(b) = 9, g(c) = 9, g(d) = 11, g(e) = 9, g(f) = 7. Find f 1 ({7, 9}) = f 1 ({12, 13}) = f 1 ({11, 12}) = EXAMPLE 17. f : R R is defined by f(x) = 3x + 4. Let W = {x R x > 0}. Find f 1 (W ).

7 c Dr Oksana Shatalov, Fall EXAMPLE 18. f : Z Z is defined by Compute (a) f 1 ({6, 7}) = f(n) = { n/2 if n E, n + 1 if n O. (b) f 1 (O) PROPOSITION 19. Let A and B be nonempty sets and f : A B be a function. If W and V are subsets of B then (a) f 1 (W V ) = f 1 (W ) f 1 (V ). (b) f 1 (W V ) = f 1 (W ) f 1 (V ).

8 c Dr Oksana Shatalov, Fall Section 3.2 Surjective and Injective Functions Surjective functions ( onto ) DEFINITION 20. Let f : A B be a function. Then f is surjective (or a surjection) if the image of f coincides with its codomain, i.e. Imf = B. Note: surjection is also called onto. Proving surjection: We know that for all f : A B: Thus, to show that f : A B is a surjection it is sufficient to prove that In other words, to prove that f : A B is a surjective function it is sufficient to show that EXAMPLE 21. Determine which of the following functions are surjective. Give a formal proof of your answer. (a) Identity function (b) f : R R, f(x) = x 5.

9 c Dr Oksana Shatalov, Fall (c) f : Z Z, f(n) = { n 2 if n E, 2n 1 if n O. (d) f : Z Z, f(n) = { n + 1 if n E, n 3 if n O.

10 c Dr Oksana Shatalov, Fall Injective functions ( one to one ) DEFINITION 22. Let f : A B be a function. Then f is injective (or an injection) if whenever a 1, a 2 A and a 1 a 2, we have f(a 1 ) f(a 2 ). Note: surjection is also called onto. Using diagram: EXAMPLE 23. Given A = {1, 2, 3} and B = {3, 4, 5}. (a) Write out an injective function with domain A and codomain B. Justify your answer. (b) Write out a non injective function with domain A and codomain B. Justify your answer. Proving injection: Let P (a 1, a 2 ) : a 1 a 2 and Q(a 1, a 2 ) : f(a 1 ) f(a 2 ). Then by definition f is injective if. Using contrapositive, we have. EXAMPLE 24. Determine which of the following functions are injective. Give a formal proof of your answer. (a) f : R R, f(x) = 5 x.

11 c Dr Oksana Shatalov, Fall (b) f : Z Z, f(n) = { n/2 if n E, 2n if n O. (c) f : R R, f(x) = 3x 5 + 5x 3 + 2x

12 c Dr Oksana Shatalov, Fall Bijective functions DEFINITION 25. A function that is both surjective and injective is called bijective (or bijection.) EXAMPLE 26. Determine which of the following functions are bijective. Give a formal proof of your answer. (a) f : R R, f(x) = x 3. (b) f : R R, f(x) = x 2. (c) f : R R, f(x) = ax + b, where a, b R.

13 c Dr Oksana Shatalov, Fall EXAMPLE 27. Show that f : R {2} R {3} defined by f(x) = 3x x 2 is bijective. Permutations DEFINITION 28. Let A be any set. A bijection f : A A is called a permutation of A. EXAMPLE 29. Let A = {1, 2, 3} and f : A A is defined by f(1) = 3, f(2) = 1, f(3) = 2.

14 c Dr Oksana Shatalov, Fall EXAMPLE 30. Identity function is a permutation. EXAMPLE 31. f : R R is defined by f(x) = x 3. REMARK 32. If A = n then there exists n! different permutations of A. 3.3 Composition and Invertible Functions DEFINITION 33. Let A and B be nonempty sets. We define the set of all functions from A to B. If A = B, we simply write F (A). Composition of Functions F (A, B) = DEFINITION 34. Let A, B, and C be nonempty sets, and let f F (A, B), g F (B, C). We define a function gf F (A, C), called the composition of f and g, by gf(a) =

15 c Dr Oksana Shatalov, Fall EXAMPLE 35. Let A = {1, 2, 3, 4}, B = {a, b, c, d}, C = {r, s, t, u, v} and define the functions f F (A, B), g F (B, C) by Find gf. What is about fg? f = {(1, b), (2, d), (3, a), (4, a)}, g = {(a, u), (b, r), (c, r), (d, s)}. EXAMPLE 36. Let f, g R be defined by f(x) = e x and g(x) = x sin x. Find fg and gf.

16 c Dr Oksana Shatalov, Fall EXAMPLE 37. Let A = R {0} and f F (A) is defined by f(x) = 1 1 x fff. for all x R. Determine EXAMPLE 38. Let f, g F (Z) be defined by { n + 4, if n E f(n) = 2n 3, if n O g(n) = { 2n 4, if n E (n 1)/2, if n O Find gf and fg.

17 c Dr Oksana Shatalov, Fall PROPOSITION 39. Let f F (A, B) and g F (B, C). Then i. If f and g are surjections, then gf is also a surjection. Proof. ii. If f and g are injections, then gf is also an injection. Proof. COROLLARY 40. If f and g are bijections, then gf is also a bijection.

18 c Dr Oksana Shatalov, Fall PROPOSITION 41. Let f F (A, B). Then fi A = f and i B f = f. Inverse Functions DEFINITION 42. Let f F (A, B). Then f is invertible if there is a function f 1 F (B, A) such that f 1 f = i A and ff 1 = i B. If f 1 exists then it is called the inverse function of f. REMARK 43. f is invertible if and only if f 1 is invertible.

19 c Dr Oksana Shatalov, Fall PROPOSITION 44. The inverse function is unique. Proof. EXAMPLE 45. The function f : R {2} R {3} defined by f(x) = 3x is known to be bijective x 2 (see Example 8, Section 3.2). Determine the inverse f 1 (x), where x R {3}.

20 c Dr Oksana Shatalov, Fall REMARK 46. Finding the inverse of a bijective function is not always possible by algebraic manipulations. For example, if f(x) = e x then f 1 (x) = The function f(x) = 3x 5 +5x 3 +2x+2014 is known to be bijective, but there is no way to find expression for its inverse. THEOREM 47. Let A and B be sets, and let f F (A, B). Then f is invertible if and only if f is bijective.

### 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

### 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

### Functions If (x, y 1 ), (x, y 2 ) S, then y 1 = y 2

Functions 4-3-2008 Definition. A function f from a set X to a set Y is a subset S of the product X Y such that if (, y 1 ), (, y 2 ) S, then y 1 = y 2. Instead of writing (, y) S, you usually write f()

### 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.

### 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

### Chapter 5: The Integers

c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition

### Cartesian 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

### CS100: DISCRETE STRUCTURES

1 CS100: DISCRETE STRUCTURES Computer Science Department Lecture 2: Functions, Sequences, and Sums Ch2.3, Ch2.4 2.3 Function introduction : 2 v Function: task, subroutine, procedure, method, mapping, v

### MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.

MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if

### 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

### 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,

### 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

### Chapter-2 Relations and Functions. Miscellaneous

1 Chapter-2 Relations and Functions Miscellaneous Question 1: The relation f is defined by The relation g is defined by Show that f is a function and g is not a function. The relation f is defined as It

### 4.1 Real-valued functions of a real variable

Chapter 4 Functions When introducing relations from a set A to a set B we drew an analogy with co-ordinates in the x-y plane. Instead of coming from R, the first component of an ordered pair comes from

### Section 7.2: One-to-One, Onto and Inverse Functions

Section 7.2: One-to-One, Onto and Inverse Functions In this section we shall developed the elementary notions of one-to-one, onto and inverse functions, similar to that developed in a basic algebra course.

### 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

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### 1 Take-home exam and final exam study guide

Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number

### COMPOSITION OF FUNCTIONS

COMPOSITION OF FUNCTIONS INTERMEDIATE GROUP - MAY 21, 2017 Finishing Up Last Week Problem 1. (Challenge) Consider the set Z 5 = {congruence classes of integers mod 5} (1) List the elements of Z 5. (2)

### FREE Download Study Package from website: &

SHORT REVISION (FUNCTIONS) THINGS TO REMEMBER :. GENERAL DEFINITION : If to every value (Considered as real unless otherwise stated) of a variable which belongs to some collection (Set) E there corresponds

More Books At www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com

### 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

### SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY COIMBATORE Discrete Mathematics Functions

1 SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY COIMBATORE 641010 Discrete Mathematics Functions 1. Define function. Let X and Y be any two sets. A relation f from X to Y is called a function if for every x

### MAT3500/ Mandatory assignment 2013 Solutions

MAT3500/4500 - Mandatory assignment 2013 s Problem 1 Let X be a topological space, A and B be subsets of X. Recall the definition of the boundary Bd A of a set A. Prove that Bd (A B) (Bd A) (Bd B). Discuss

### Continuity. Chapter 4

Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

### 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,

### 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.

### and problem sheet 6

2-28 and 5-5 problem sheet 6 Solutions to the following seven exercises and optional bonus problem are to be submitted through gradescope by 2:0AM on Thursday 9th October 207. There are also some practice

### cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 1 Course Web Page www3.cs.stonybrook.edu/ cse303 The webpage contains: lectures notes slides; very detailed solutions to

### A function is a special kind of relation. More precisely... A function f from A to B is a relation on A B such that. f (x) = y

Functions A function is a special kind of relation. More precisely... A function f from A to B is a relation on A B such that for all x A, there is exactly one y B s.t. (x, y) f. The set A is called the

### Solutions to Homework Set 1

Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement

### 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

### MATH 436 Notes: Homomorphisms.

MATH 436 Notes: Homomorphisms. Jonathan Pakianathan September 23, 2003 1 Homomorphisms Definition 1.1. Given monoids M 1 and M 2, we say that f : M 1 M 2 is a homomorphism if (A) f(ab) = f(a)f(b) for all

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### Fibers, Surjective Functions, and Quotient Groups

Fibers, Surjective Functions, and Quotient Groups 11/01/06 Radford Let f : X Y be a function. For a subset Z of X the subset f(z) = {f(z) z Z} of Y is the image of Z under f. For a subset W of Y the subset

### 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.

### 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

### BRAIN TEASURES FUNCTION BY ABHIJIT KUMAR JHA EXERCISE I. log 5. (ii) f (x) = log 7. (iv) f (x) = 2 x. (x) f (x) = (xii) f (x) =

EXERCISE I Q. Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f () = cos 6 (ii) f () = log

### 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

### 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

### 5. Some theorems on continuous functions

5. Some theorems on continuous functions The results of section 3 were largely concerned with continuity of functions at a single point (usually called x 0 ). In this section, we present some consequences

### Review of Functions. Functions. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Functions Current Semester 1 / 12

Review of Functions Functions Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Functions Current Semester 1 / 12 Introduction Students are expected to know the following concepts about functions:

### 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

### Ch 7 Summary - POLYNOMIAL FUNCTIONS

Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)

### MATH 409 Advanced Calculus I Lecture 11: More on continuous functions.

MATH 409 Advanced Calculus I Lecture 11: More on continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if for any ε > 0 there

### 4. We accept without proofs that the following functions are differentiable: (e x ) = e x, sin x = cos x, cos x = sin x, log (x) = 1 sin x

4 We accept without proofs that the following functions are differentiable: (e x ) = e x, sin x = cos x, cos x = sin x, log (x) = 1 sin x x, x > 0 Since tan x = cos x, from the quotient rule, tan x = sin

### Differentiation. f(x + h) f(x) Lh = L.

Analysis in R n Math 204, Section 30 Winter Quarter 2008 Paul Sally, e-mail: sally@math.uchicago.edu John Boller, e-mail: boller@math.uchicago.edu website: http://www.math.uchicago.edu/ boller/m203 Differentiation

### D-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 3. Modular arithmetic, quotients, product groups

D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 3 Modular arithmetic, quotients, product groups 1. Show that the functions f = 1/x, g = (x 1)/x generate a group of functions, the law of composition

### 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

### CHAPTER 1: RELATIONS AND FUNCTIONS

CHAPTER 1: RELATIONS AND FUNCTIONS Previous Years Board Exam (Important Questions & Answers) 1. If f(x) = x + 7 and g(x) = x 7, x R, find ( fog) (7) Given f(x) = x + 7 and g(x) = x 7, x R fog(x) = f(g(x))

### 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

### Rings and Fields Theorems

Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

### GALOIS CATEGORIES MELISSA LYNN

GALOIS CATEGORIES MELISSA LYNN Abstract. In abstract algebra, we considered finite Galois extensions of fields with their Galois groups. Here, we noticed a correspondence between the intermediate fields

### 2. Isometries and Rigid Motions of the Real Line

21 2. Isometries and Rigid Motions of the Real Line Suppose two metric spaces have different names but are essentially the same geometrically. Then we need a way of relating the two spaces. Similarly,

### ACCRS/QUALITY CORE CORRELATION DOCUMENT: ALGEBRA I

ACCRS/QUALITY CORE CORRELATION DOCUMENT: ALGEBRA I Revised March 25, 2013 Extend the properties of exponents to rational exponents. 1. [N-RN1] Explain how the definition of the meaning of rational exponents

### MATHEMATICS HIGHER SECONDARY FIRST YEAR VOLUME II REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE

MATHEMATICS HIGHER SECONDARY FIRST YEAR VOLUME II REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE Untouchability is a sin Untouchability is a crime Untouchability is inhuman

### 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

### Show Your Work! Point values are in square brackets. There are 35 points possible. Tables of tautologies and contradictions are on the last page.

Formal Methods Midterm 1, Spring, 2007 Name Show Your Work! Point values are in square brackets. There are 35 points possible. Tables of tautologies and contradictions are on the last page. 1. Use truth

### 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

### Discrete Mathematics. Thomas Goller. January 2013

Discrete Mathematics Thomas Goller January 2013 Contents 1 Mathematics 1 1.1 Axioms..................................... 1 1.2 Definitions................................... 2 1.3 Theorems...................................

### 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

### Supplementary Material for MTH 299 Online Edition

Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think

### SECTION 1.8 : x = f LEARNING OBJECTIVES

SECTION 1.8 : x = f (Section 1.8: x = f ( y) ( y)) 1.8.1 LEARNING OBJECTIVES Know how to graph equations of the form x = f ( y). Compare these graphs with graphs of equations of the form y = f ( x). Recognize

### Functional Equations

Functional Equations Henry Liu, 22 December 2004 henryliu@memphis.edu Introduction This is a brief set of notes on functional equations. It is one of the harder and less popular areas among Olympiad problems,

### MAS114: Numbers and Groups, Semester 2

MAS114: Numbers and Groups, Semester 2 Module Information Contact details. Dr Sam Marsh (Room G9, Hick Building). Office hour. FILL THIS IN HERE: Friday 12, or see the course website. No appointment necessary,

### How many elements does B have to have? At least one, since A is not empty. If

15. Injective surjective and bijective The notion of an invertible function is very important and we would like to break up the property of being invertible into pieces. Definition 15.1. Let f : A B be

### Real Analysis. Jesse Peterson

Real Analysis Jesse Peterson February 1, 2017 2 Contents 1 Preliminaries 7 1.1 Sets.................................. 7 1.1.1 Countability......................... 8 1.1.2 Transfinite induction.....................

### INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS

INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS JAMES P. KELLY AND JONATHAN MEDDAUGH Abstract. In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous

### ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018)

ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018) These are errata for the Third Edition of the book. Errata from previous editions have been

### f(x) f(z) c x z > 0 1

INVERSE AND IMPLICIT FUNCTION THEOREMS I use df x for the linear transformation that is the differential of f at x.. INVERSE FUNCTION THEOREM Definition. Suppose S R n is open, a S, and f : S R n is a

### Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible.

Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.3.5, 4.3.7, 4.3.8, 4.3.9,

### Set Basics. P. Danziger

- 5 Set Basics P. Danziger 1 Set Basics Definition 1 (Cantor) A set is a collection into a whole of definite and separate objects of our intuition or thought. (Russell) Such that given an object x and

### MS 2001: Test 1 B Solutions

MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question

### BASIC GROUP THEORY : G G G,

BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e

### 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

### Math 3140 Fall 2012 Assignment #3

Math 3140 Fall 2012 Assignment #3 Due Fri., Sept. 21. Remember to cite your sources, including the people you talk to. My solutions will repeatedly use the following proposition from class: Proposition

### FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen

### NATIONAL OPEN UNIVERSITY OF NIGERIA COURSE CODE : MTH 211 COURSE TITLE: SET THEORY AND ABSTRACT ALGEBRA

NATIONAL OPEN UNIVERSITY OF NIGERIA COURSE CODE : MTH 211 COURSE TITLE: SET THEORY AND ABSTRACT ALGEBRA COURSE GUIDE COURSE GUIDE MTH 211 Adapted from Course Team Indira Gandhi National Open University

### 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

### 3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first

### Basics on Sets and Functions

HISTORY OF MATHEMATICS Spring 2005 Basics on Sets and Functions Introduction to the work of Georg Cantor #10 in the series. Some dictionaries define a set as a collection or aggregate of objects. However,

### Test 2 Review Math 1111 College Algebra

Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.

### Chapter 2. Relations, Functions, Partial Functions

Chapter 2 Relations, Functions, Partial Functions 2.1 What is a Function? Roughly speaking, a function, f, is a rule or mechanism, which takes input values in some input domain, say X, and produces output

### arxiv: v1 [cs.pl] 19 May 2016

arxiv:1605.05858v1 [cs.pl] 19 May 2016 Domain Theory: An Introduction Robert Cartwright Rice University Rebecca Parsons ThoughtWorks, Inc. Moez AbdelGawad SRTA-City Hunan University This monograph is an

### Inverse Functions. One-to-one. Horizontal line test. Onto

Inverse Functions One-to-one Suppose f : A B is a function. We call f one-to-one if every distinct pair of objects in A is assigned to a distinct pair of objects in B. In other words, each object of the

### Primitive recursive functions: decidability problems

Primitive recursive functions: decidability problems Armando B. Matos October 24, 2014 Abstract Although every primitive recursive (PR) function is total, many problems related to PR functions are undecidable.

### ISOMORPHISMS KEITH CONRAD

ISOMORPHISMS KEITH CONRAD 1. Introduction Groups that are not literally the same may be structurally the same. An example of this idea from high school math is the relation between multiplication and addition

### Abstract Algebra I. Randall R. Holmes Auburn University. Copyright c 2012 by Randall R. Holmes Last revision: November 11, 2016

Abstract Algebra I Randall R. Holmes Auburn University Copyright c 2012 by Randall R. Holmes Last revision: November 11, 2016 This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives

### MATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM

MATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM (FP1) The exclusive or operation, denoted by and sometimes known as XOR, is defined so that P Q is true iff P is true or Q is true, but not both. Prove (through

### Descending chains and antichains of the unary, linear, and monotone subfunction relations

Descending chains and antichains of the unary, linear, and monotone subfunction relations Erkko Lehtonen November 21, 2005 Abstract The C-subfunction relations on the set of functions on a finite base

### Walker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015

Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of

### 1. General Vector Spaces

1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

### 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

### Groups. Contents of the lecture. Sergei Silvestrov. Spring term 2011, Lecture 8

Groups Sergei Silvestrov Spring term 2011, Lecture 8 Contents of the lecture Binary operations and binary structures. Groups - a special important type of binary structures. Isomorphisms of binary structures.

### Functions. Chapter Continuous Functions

Chapter 3 Functions 3.1 Continuous Functions A function f is determined by the domain of f: dom(f) R, the set on which f is defined, and the rule specifying the value f(x) of f at each x dom(f). If f is

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### = x iy. Also the. DEFINITION #1. If z=x+iy, then the complex conjugate of z is given by magnitude or absolute value of z is z =.

Handout # 4 COMPLEX FUNCTIONS OF A COMPLEX VARIABLE Prof. Moseley Chap. 3 To solve z 2 + 1 = 0 we "invent" the number i with the defining property i 2 = ) 1. We then define the set of complex numbers as