# Applications of Methods of Proof

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are analogous to the operations,, and, respectively, that were defined for propositions. Indeed, each set operation was defined in terms of the corresponding operator from logic. We will discuss these operations in some detail in this section and learn methods to prove some of their basic properties. Recall that in any discussion about sets and set operations there must be a set, called a universal set, that contains all others sets to be considered. This term is a bit of a misnomer: logic prohibits the existence of a set of all sets, so that there is no one set that is universal in this sense. Thus the choice of a universal set will depend on the problem at hand, but even then it will in no way be unique. As a rule we usually choose one that is minimal to suit our needs. For example, if a discussion involves the sets {1, 2, 3, 4} and {2, 4, 6, 8, 10}, we could consider our universe to be the set of natural numbers or the set of integers. On the other hand, we might be able to restrict it to the set of numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. We now restate the operations of set theory using the formal language of logic Equality and Containment. Definition Sets A and B are equal, denoted A = B, if x[x A x B] Note: This is equivalent to x[(x A x B) (x B x A)]. Definition Set A is contained in set B (or A is a subset of B), denoted A B,if x[x A x B]. 96

2 1. SET OPERATIONS 97 The above note shows that A = B iff A B and B A Union and Intersection. Definitions The union of A and B, The intersection of A and B, A B = {x (x A) (x B)}. A B = {x (x A) (x B)}. If A B =, then A and B are said to be disjoint Complement. Definition The complement of A A = {x U (x A)} = {x U x A}. There are several common notations used for the complement of a set. For example, A c is often used to denote the complement of A. You may find it easier to type A c than A, and you may use this notation in your homework Difference. Definition The difference of A and B, or the complement of B relative to A, A B = A B. Definition The symmetric difference of A and B, A B = (A B) (B A) = (A B) (A B). The difference and symmetric difference of two sets are new operations, which were not defined in Module 1.1. Notice that B does not have to be a subset of A for

3 1. SET OPERATIONS 98 the difference to be defined. This gives us another way to represent the complement of a set A; namely, A = U A, where U is the universal set. The definition of the difference of two sets A and B in some universal set, U, is equivalent to A B = {x U (x A) (x B)}. Many authors use the notation A \ B for the difference A B. The symmetric difference of two sets corresponds to the logical operation, the exclusive or. The definition of the symmetric difference of two sets A and B in some universal set, U, is equivalent to A B = {x U [(x A) (x B)] [ (x A) (x B)]} Product. Definition The (Cartesian) Product of two sets, A and B, is denoted A B and is defined by 1.7. Power Set. A B = {(a, b) a A b B} Definition Let S be a set. The power set of S, denoted P(S) is defined to be the set of all subsets of S. Keep in mind the power set is a set where all the elements are actually sets and the power set should include the empty set and itself as one of its elements Examples. Example Assume: U = {a, b, c, d, e, f, g, h}, A = {a, b, c, d, e}, B = {c, d, e, f}, and C = {a, b, c, g, h}. Then (a) A B = {a, b, c, d, e, f} (b) A B = {c, d, e} (c) A = {f, g, h} (d) B = {a, b, g, h} (e) A B = {a, b} (f) B A = {f} (g) A B = {a, b, f}

4 1. SET OPERATIONS 99 (h) (A B) C = {a, b, c} (i) A B = {(a, c), (a, d), (a, e), (a, f), (b, c), (b, d), (b, e), (b, f), (c, c), (c, d), (c, e), (c, f), (d, c), (d, d), (d, e), (d, f), (e, c), (e, d), (e, e), (e, f)} (j) P(A) = {, {a}, {b}, {c}, {d}, {e}, {a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}, {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}, {b, c, d, e}, {a, b, c, d, e}} (k) P(A) = 32 Exercise Use the sets given in Example to find (1) B A (2) P(B) (3) P(U) Example Let the universal set be U = Z + the set of all positive integers, let P be the set of all prime (positive) integers, and let E be the set of all positive even integers. Then (a) P E = {n Z + n is prime or even}, (b) P E = {2}, (c) P is the set of all positive composite integers, (d) E is the set of all positive odd integers, {2n + 1 n N}, (e) P E is the set of all positive odd prime numbers (all prime numbers except 2), (f) E P = {4, 6, 8, 10,... } = {2n n Z + n 2}, (g) E P = {n Z + (n is prime or even) n 2} Exercise (1) If A = n and B = m, how many elements are in A B? (2) If S is a set with S = n, what is P(S)? Exercise Does A B = B A? Prove your answer Venn Diagrams. A Venn Diagram is a useful geometric visualization tool when dealing with three or fewer sets. The Venn Diagram is generally set up as follows: The Universe U is the rectangular box. A set are represented by a circle and its interior. In the absence of specific knowledge about the relationships among the sets being represented, the most generic relationships should be depicted.

5 1. SET OPERATIONS 100 Venn Diagrams can be very helpful in visualizing set operations when you are dealing with three or fewer sets (not including the universal set). They tend not to be as useful, however, when considering more than three sets. Although Venn diagrams may be helpful in visualizing sets and set operations, they will not be used for proving set theoretic identities Examples. Example The following Venn Diagrams illustrate generic relationships between two and three sets, respectively. U U A B A B C Example This Venn Diagram represents the difference A B (the shaded region). U A A - B B The figures in the examples above show the way you might draw the Venn diagram if you aren t given any particular relations among the sets. On the other hand, if you knew, for example, that A B, then you would draw the set A inside of B.

6 1. SET OPERATIONS Set Identities. Example Prove that the complement of the union is the intersection of the complements: A B = A B. Proof 1. One way to show the two sets are equal is to use the fact that A B = A B iff A B A B and A B A B. Step 1. Show A B A B. Assume x is an arbitrary element of A B (and show x A B). Since x A B, x A B. This means x A and x B (De Morgan s Law). Hence x A B. Thus, by Universal Generalization, so that, by definition, x[x (A B) x (A B)] A B A B. Step 2. Show A B A B. Suppose x is an arbitrary element of A B. Then x A and x B. Therefore, x A B (De Morgan s Law). This shows x A B. Thus, by Universal Generalization, so that, by definition, x[x (A B) x (A B)] A B A B. Proof 2. The following is a second proof of the same result, which emphasizes more clearly the role of the definitions and laws of logic. We will show x[x A B x A B].

7 1. SET OPERATIONS 102 Assertion x: x A B x [A B] [x A B] [(x A) (x B)] (x A) (x B) (x A) (x B) x A B Reason Definition of complement Definition of Definition of union De Morgan s Law Definition of complement Definition of intersection Hence x[x A B x A B] is a tautology. (In practice we usually omit the formality of writing x in the initial line of the proof and assume that x is an arbitrary element of the universe of discourse.) Proof 3. A third way to prove this identity is to build a membership table for the sets A B and A B, and show the membership relations for the two sets are the same. The 1 s represent membership in a set and the 0 s represent nonmembership. A B A B A B A B A B Compare this table to the truth table for the proof of De Morgan s Law: (p q) ( p q) A set identity is an equation involving sets and set operations that is true for all possible choices of sets represented by symbols in the identity. These are analgous to identities such as (a + b)(a b) = a 2 b 2

8 that you encounter in an elementary algebra course. 1. SET OPERATIONS 103 There are various ways to prove an identity, and three methods are covered here. This is a good place to be reminded that when you are proving an identity, you must show that it holds in all possible cases. Remember, giving an example does not prove an identity. On the other hand, if you are trying to show that an expression is not an identity, then you need only provide one counterexample. (Recall the negation of xp (x) is x P (x)). Proof 1 establishes equality by showing each set is a subset of the other. This method can be used in just about any situation. Notice that in Proof 1 we start with the assumption, x is in A B, where x is otherwise an arbitrary element in some universal set. If we can show that x must then be in A B, then we will have established x, [x A B] [x A B]. That is, the modus operandi is to prove the implications hold for an arbitrary element x of the universe, concluding, by Universal Generalization, that the implications hold for all such x. Notice the way De Morgan s Laws are used here. For example, in the first part of Proof 1, we are given that x (A B). This means [x (A B)] [(x A) (x B)] [(x A) (x B)]. Proof 2 more clearly exposes the role of De Morgan s Laws. Here we prove the identity by using propositional equivalences in conjunction with Universal Generalization. When using this method, as well as any other, you must be careful to provide reasons. Proof 3 provides a nice alternative when the identity only involves a small number of sets. Here we show two sets are equal by building a member table for the sets. The member table has a 1 to represent the case in which an element is a member of the set and a 0 to represent the case when it is not. The set operations correspond to a logical connective and one can build up to the column for the set desired. You will have proved equality if you demonstrate that the two columns for the sets in question have the exact same entries. Notice that all possible membership relations of an element in the universal set for the sets A and B are listed in the first two columns of the membership table. For example, if an element is in both A and B in our example, then it satisfies the conditions in the first row of the table. Such an element ends up in neither of the two sets A B nor A B. This is very straight forward method to use for proving a set identity. It may also be used to prove containment. If you are only trying to show the containment M N, you would build the membership table for M and N as above. Then you would look in every row where M has a 1 to see that

9 1. SET OPERATIONS 104 N also has a 1. However, you will see examples in later modules where a membership table cannot be created. It is not always possible to represent all the different possibilities with a membership table. Example Prove the identity (A B) C = (A C) (B C) Proof 1. Suppose x is an arbitrary element of the universe. Assertion x (A B) C [x (A B)] [x C] [(x A) (x B)] [x C] [(x A) (x C)] [(x B) (x C)] [x (A C)] [(x (B C)] x [(A C) (B C)] Reason definition of intersection definition of union distributive law of and over or definition of intersection definition of union Proof 2. Build a membership table: A B C A B (A C) (B C) (A B) C (A C) (B C) Since the columns corresponding to (A B) C and (A C) (B C) are identical, the two sets are equal.

10 Example Prove (A B) C A (B C). 1. SET OPERATIONS 105 Proof. Consider the membership table: A B C A B B C (A B) C A (B C) Notice the only 1 in the column for (A B) C is the fourth row. The entry in the same row in the column for A (B C) is also a 1, so (A B) C A (B C). Exercise Prove the identity A B = A B using the method of Proof 2 in Example Exercise Prove the identity A B = A B using the method of Proof 3 in Example Exercise Prove the identity (A B) C = (A C) (B C) using the method of Proof 1 in Example Exercise Prove the identity (A B) C = (A C) (B C) using the method of Proof 2 in Example Exercise Prove the identity (A B) C = (A C) (B C) using the method of Proof 3 in Example Union and Intersection of Indexed Collections. Definition The the union and intersection of an indexed collection of sets {A 1, A 2, A 3,..., A n } can be written as n A i = A 1 A 2 A 3 A n

11 1. SET OPERATIONS 106 and respectively. n A i = A 1 A 2 A 3 A n, Infinite Unions and Intersections. Definition The union and intersection of an indexed collection of infinitely many sets {A 1, A 2, A 3,..., } can be written as and A i = {a a A i for some i in Z + } A i = {a a A i for all i in Z + } If you have a collection of more than two sets, you can define the intersection and the union of the sets as above. (Since the operations are associative, it isn t necessary to clutter the picture with parentheses.) The notation is similar to the Σ notation used for summations. The subscript is called an index and the collection of sets is said to be indexed by the set of indices. In the example, the collection of sets is {A 1, A 2,..., A n }, and the set of indices is the set {1, 2,..., n}. There is no requirement that sets with different indices be different. In fact, they could all be the same set. This convention is very useful when each of the sets in the collection is naturally described in terms of the index (usually a number) it has been assigned. An equivalent definition of the union and intersection of an indexed collection of sets is as follows: and n A i = {x i {1, 2,..., n} such that x A i } n A i = {x i {1, 2,..., n}, x A i }.

12 1. SET OPERATIONS 107 Another standard notation for unions over collections of indices is A i = A i. i Z + More generally, if I is any set of indices, we can define A i = {x i I such that x A i }. i I Example Example Let A i = [i, i + 1), where i is a positive integer. Then n A i = [1, n + 1), and n A i =, if n > 1. A i = [1, ) A i = This is an example of a collection of subsets of the real numbers that is naturally indexed. If A i = [i, i + 1), then A 1 = [1, 2), A 2 = [2, 3), A 3 = [3, 4), etc. It may help when dealing with an indexed collection of sets to explicitly write out a few of the sets as we have done here. Example Suppose C i = {i 2, i 1, i, i + 1, i + 2}, where i denotes an arbitrary natural number. Then C 0 = { 2, 1, 0, 1, 2}, C 1 = { 1, 0, 1, 2, 3}, C 2 = {0, 1, 2, 3, 4}, n C i = { 2, 1, 0, 1,..., n, n + 1, n + 2} i=0 4 C i = {2} i=0

13 n C i = if n > 4. i=0 C i = { 2, 1, 0, 1, 2, 3,... } i=0 C i = i=0 1. SET OPERATIONS 108 Exercise For each positive integer k, let A k = {kn n Z}. For example, Find A 1 = {n n Z} = Z A 2 = {2n n Z} = {..., 2, 0, 2, 4, 6,...} A 3 = {3n n Z} = {..., 3, 0, 3, 6, 9,...} A k k=1 m A k, where m is an arbitrary positive integer. k=1 Exercise Use the definition for A k in exercise to answer the following questions. (1) (2) A i A i Computer Representation of a Set. Here is a method for storing subsets of a given, finite universal set: Order the elements of the universal set and then assign a bit number to each subset A as follows. A bit is 1 if the element corresponding to the position of the bit in the universal set is in A, and 0 otherwise. Example Suppose U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with the obvious ordering. Then The bit string corresponding to A = {2, 4, 6, 8, 10} is The bit string corresponding to B = {1, 2, 3, 4} is

14 1. SET OPERATIONS 109 There are many ways sets may be represented and stored in a computer. One such method is presented here. Notice that this method depends not only on the universal set, but on the order of the universal set as well. If we rearrange the order of the universal set given in Example to U = {10, 9, 8, 7, 6, 5, 4, 3, 2, 1}, then the bit string corresponding to the subset {1, 2, 3, 4} would be The beauty of this representation is that set operations on subsets of U can be carried out formally using the corresponding bitstring operations on the bit strings representing the individual sets. Example For the sets A, B U in Example : A = ( ) = = {1, 3, 5, 7, 9} A B = = = {1, 2, 3, 4, 6, 8, 10} A B = = = {2, 4} A B = = = {1, 3, 6, 8, 10} Exercise Let U = {a, b, c, d, e, f, g, h, i, j} with the given alphabetical order. Let A = {a, e, i}, B = {a, b, d, e, g, h, j}, and C = {a, c, e, g, i}. (1) Write out the bit string representations for A, B, and C. (2) Use these representations to find (a) C

15 (b) A B (c) A B C (d) B C 1. SET OPERATIONS 110

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### Introduction Russell s Paradox Basic Set Theory Operations on Sets. 6. Sets. Terence Sim

6. Sets Terence Sim 6.1. Introduction A set is a Many that allows itself to be thought of as a One. Georg Cantor Reading Section 6.1 6.3 of Epp. Section 3.1 3.4 of Campbell. Familiar concepts Sets can

### Sets. 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

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical

Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### Notes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.

Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### 2.1 Sets, power sets. Cartesian Products.

Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects. - used to group objects together, - often the objects with similar properties This description of a set (without

### The Language of Mathematics

CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,

### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

### Math 117 Chapter 7 Sets and Probability

Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a well-defined collection of specific objects. Each item in the set is called an element or a member. Curly

### The set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;

Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method

### Sets 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.

### CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)

### 2.1.1 Examples of Sets and their Elements

Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define

### MAT2400 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

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### LESSON SUMMARY. Set Operations and Venn Diagrams

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Three: Set Theory Lesson 4 Set Operations and Venn Diagrams Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volumes 1 and 2. (Some helpful exercises

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### Sets, Relations and Functions

Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### CARDINALITY, 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

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### SETS. 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

### What is a set? Sets. Specifying a Set. Notes. The Universal Set. Specifying a Set 10/29/13

What is a set? Sets CS 231 Dianna Xu set is a group of objects People: {lice, ob, Clara} Colors of a rainbow: {red, orange, yellow, green, blue, purple} States in the S: {labama, laska, Virginia, } ll

### A set is an unordered collection of objects.

Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain

### CS 341 Homework 9 Languages That Are and Are Not Regular

CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w

### Lecture 1. Basic Concepts of Set Theory, Functions and Relations

September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2

### Introducing 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

### 4.1. Sets. Introduction. Prerequisites. Learning Outcomes. Learning Style

ets 4.1 Introduction If we can identify a property which is common to several objects, it is often useful to group them together. uch a grouping is called a set. Engineers for example, may wish to study

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Introduction to Proofs

Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 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

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### Chapter 1. Logic and Proof

Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

### Introduction to mathematical arguments

Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### 2.1 Symbols and Terminology

2.1 Symbols and Terminology Definitions: set is a collection of objects. The objects belonging to the set are called elements, ormembers, oftheset. Sets can be designated in one of three different ways:

### Sets and Logic. Chapter Sets

Chapter 2 Sets and Logic This chapter introduces sets. In it we study the structure on subsets of a set, operations on subsets, the relations of inclusion and equality on sets, and the close connection

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Discrete Mathematics

Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can

### Definition 14 A set is an unordered collection of elements or objects.

Chapter 4 Set Theory Definition 14 A set is an unordered collection of elements or objects. Primitive Notation EXAMPLE {1, 2, 3} is a set containing 3 elements: 1, 2, and 3. EXAMPLE {1, 2, 3} = {3, 2,

### Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the

### 2.1 The Algebra of Sets

Chapter 2 Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations

### Math 166 - Week in Review #4. A proposition, or statement, is a declarative sentence that can be classified as either true or false, but not both.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Week in Review #4 Sections A.1 and A.2 - Propositions, Connectives, and Truth Tables A proposition, or statement, is a declarative sentence that

### 31 is a prime number is a mathematical statement (which happens to be true).

Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### Mathematical Induction

Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### It 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

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### This 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.

### 1.1 Logical Form and Logical Equivalence 1

Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### the lemma. Keep in mind the following facts about regular languages:

CPS 2: Discrete Mathematics Instructor: Bruce Maggs Assignment Due: Wednesday September 2, 27 A Tool for Proving Irregularity (25 points) When proving that a language isn t regular, a tool that is often

### We 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

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### Lecture 12. More algebra: Groups, semigroups, monoids, strings, concatenation.

V. Borschev and B. Partee, November 1, 2001 p. 1 Lecture 12. More algebra: Groups, semigroups, monoids, strings, concatenation. CONTENTS 1. Properties of operations and special elements...1 1.1. Properties

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### Solving and Graphing Inequalities

Algebra I Pd Basic Inequalities 3A What is the answer to the following questions? Solving and Graphing Inequalities We know 4 is greater than 3, so is 5, so is 6, so is 7 and 3.1 works also. So does 3.01

### The Mathematics Driving License for Computer Science- CS10410

The Mathematics Driving License for Computer Science- CS10410 Venn Diagram, Union, Intersection, Difference, Complement, Disjoint, Subset and Power Set Nitin Naik Department of Computer Science Venn-Euler

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### 2. Propositional Equivalences

2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

### Sample 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

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### WUCT121. Discrete Mathematics. Logic

WUCT121 Discrete Mathematics Logic 1. Logic 2. Predicate Logic 3. Proofs 4. Set Theory 5. Relations and Functions WUCT121 Logic 1 Section 1. Logic 1.1. Introduction. In developing a mathematical theory,

### Fundamentele Informatica II

Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### 1 Introduction to Counting

1 Introduction to Counting 1.1 Introduction In this chapter you will learn the fundamentals of enumerative combinatorics, the branch of mathematics concerned with counting. While enumeration problems can

### Finite Sets. Theorem 5.1. Two non-empty 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

### CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

### Classical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions

Classical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions Fuzzy Sets Notation Convention for Fuzzy Sets Fuzzy

### Matthias Beck Ross Geoghegan. The Art of Proof. Basic Training for Deeper Mathematics

Matthias Beck Ross Geoghegan The Art of Proof Basic Training for Deeper Mathematics ! "#\$\$%&#'!()*+!!,-''!.)-/%)/#0! 1)2#3\$4)0\$!-5!"#\$%)4#\$&*'!! 1)2#3\$4)0\$!-5!"#\$%)4#\$&*#6!7*&)0*)'! 7#0!83#0*&'*-!7\$#\$)!90&:)3'&\$;!

### The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs

The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments

### Mathematics Review for MS Finance Students

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