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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections of Rosen Introduction I We ve already implicitly dealt with sets (integers, Z; rationals (Q) etc.) but here we will develop more fully the definitions, properties and operations of sets. set is an unordered collection of (unique) objects. Sets are fundamental discrete structures that form the basis of more complex discrete structures like graphs. Contrast this definition with the one in the book (compare bag, multi-set, tuples, etc). Introduction II The objects in a set are called elements or members of a set. set is said to contain its elements. Recall the notation: for a set, an element x we write x if contains x and otherwise. Latex notation: \in, \neg\in. x

2 Terminology I Two sets, and B are equal if they contain the same elements. In this case we write = B. {2, 3, 5, 7} = {3, 2, 7, 5} since a set is unordered. lso, {2, 3, 5, 7} = {2, 2, 3, 3, 5, 7} since a set contains unique elements. However, {2, 3, 5, 7} {2, 3}. Terminology II multi-set is a set where you specify the number of occurrences of each element: {m 1 a 1, m 2 a 2,..., m r a r } is a set where m 1 occurs a 1 times, m 2 occurs a 2 times, etc. Note in CS (Databases), we distinguish: a set is w/o repetition a bag is a set with repetition Terminology III We ve already seen set builder notation: O = {x (x Z) (x = 2k for some k Z)} should be read O is the set that contains all x such that x is an integer and x is even. set is defined in intension, when you give its set builder notation. O = {x (x Z) (x 8)} set is defined in extension, when you enumerate all the elements. O = {0, 2, 6, 8}

3 Venn Diagram set can also be represented graphically using a Venn diagram. U x y z B a C Figure: Venn Diagram More Terminology & Notation I set that has no elements is referred to as the empty set or null set and is denoted. singleton set is a set that has only one element. We usually write {a}. Note the different: brackets indicate that the object is a set while a without brackets is an element. The subtle difference also exists with the empty set: that is { } The first is a set, the second is a set containing a set. More Terminology & Notation II is said to be a subset of B and we write B if and only if every element of is also an element of B. That is, we have an equivalence: B x(x x B)

4 More Terminology & Notation III Theorem For any set S, S and S S (Theorem 1, page 81.) The proof is in the book note that it is an excellent example of a vacuous proof! Latex notation: \emptyset, \subset, \subseteq. More Terminology & Notation IV set that is a subset of B is called a proper subset if B. That is, there is some element x B such that x. In this case we write B or to be even more definite we write B Let = {2}. Let B = {x (x 100) (x is prime)}. Then B. Latex notation: \subsetneq. More Terminology & Notation V Sets can be elements of other sets. {, {a}, {b}, {a, b}} and {{1}, {2}, {3}} are sets with sets for elements.

5 More Terminology & Notation VI If there are exactly n distinct elements in a set S, with n a nonnegative integer, we say that S is a finite set and the cardinality of S is n. Notationally, we write S = n set that is not finite is said to be infinite. More Terminology & Notation VII Recall the set B = {x (x 100) (x is prime)}, its cardinality is B = 25 since there are 25 primes less than 100. Note the cardinality of the empty set: = 0 The sets N, Z, Q, R are all infinite. Proving Equivalence I You may be asked to show that a set is a subset, proper subset or equal to another set. To do this, use the equivalence discussed before: B x(x x B) To show that B it is enough to show that for an arbitrary (nonspecific) element x, x implies that x is also in B. ny proof method could be used. To show that B you must show that is a subset of B just as before. But you must also show that x((x B) (x )) Finally, to show two sets equal, it is enough to show (much like an equivalence) that B and B independently.

6 Proving Equivalence II Logically speaking this is showing the following quantified statements: ( x(x a x B) ) ( x(x B x ) ) We ll see an example later. The Power Set I The power set of a set S, denoted P(S) is the set of all subsets of S. Let = {a, b, c} then the power set is P(S) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} Note that the empty set and the set itself are always elements of the power set. This follows from Theorem 1 (Rosen, p81). The Power Set II The power set is a fundamental combinatorial object useful when considering all possible combinations of elements of a set. Fact Let S be a set such that S = n, then P(S) = 2 n

7 Tuples I Sometimes we may need to consider ordered collections. The ordered n-tuple (a 1, a 2,..., a n ) is the ordered collection with the a i being the i-th element for i = 1, 2,..., n. Two ordered n-tuples are equal if and only if for each i = 1, 2,..., n, a i = b i. For n = 2, we have ordered pairs. Cartesian Products I Let and B be sets. The Cartesian product of and B denoted B, is the set of all ordered pairs (a, b) where a and b B: B = {(a, b) (a ) (b B)} The Cartesian product is also known as the cross product. subset of a Cartesian product, R B is called a relation. We will talk more about relations in the next set of slides. Cartesian Products II Note that B B unless = or B = or = B. Can you find a counter example to prove this?

8 Cartesian Products III Cartesian products can be generalized for any n-tuple. The Cartesian product of n sets, 1, 2,..., n, denoted 1 2 n is 1 2 n = {(a 1, a 2,..., a n ) a i i for i = 1, 2,..., n} Notation With Quantifiers Whenever we wrote xp (x) or xp (x), we specified the universe of discourse using explicit English language. Now we can simplify things using set notation! x R(x 2 0) x Z(x 2 = 1) Or you can mix quantifiers: a, b, c R x C(ax 2 + bx + c = 0) Set Operations Just as arithmetic operators can be used on pairs of numbers, there are operators that can act on sets to give us new sets.

9 Set Operators Union The union of two sets and B is the set that contains all elements in, B or both. We write B = {x (x ) (x B)} Latex notation: \cup. Set Operators Intersection The intersection of two sets and B is the set that contains all elements that are elements of both and B We write B = {x (x ) (x B)} Latex notation: \cap. Set Operators Venn Diagram U B Sets and B

10 Set Operators Venn Diagram : Union U B Union, B Set Operators Venn Diagram : Intersection U B Intersection, B Disjoint Sets Two sets are said to be disjoint if their intersection is the empty set: B = U B Figure: Two disjoint sets and B.

11 Set Difference The difference of sets and B, denoted by \ B (or B) is the set containing those elements that are in but not in B. U B Figure: Set Difference, \ B Latex notation: \setminus. Set Complement The complement of a set, denoted Ā, consists of all elements not in. That is, the difference of the universal set and ; U \. Ā = {x x } U Figure: Set Complement, Latex notation: \overline. Set Identities There are analogs of all the usual laws for set operations. gain, the Cheat Sheet is available on the course web page. LogicalEquivalences.pdf Let s take a quick look at this Cheat Sheet

12 Proving Set Equivalences Recall that to prove such an identity, one must show that 1. The left hand side is a subset of the right hand side. 2. The right hand side is a subset of the left hand side. 3. Then conclude that they are, in fact, equal. The book proves several of the standard set identities. We ll give a couple of different examples here. Proving Set Equivalences I Let = {x x is even} and B = {x x is a multiple of 3} and C = {x x is a multiple of 6}. Show that B = C Proof. ( B C): Let x B. Then x is a multiple of 2 and x is a multiple of 3, therefore we can write x = 2 3 k for some integer k. Thus x = 6k and so x is a multiple of 6 and x C. (C B): Let x C. Then x is a multiple of 6 and so x = 6k for some integer k. Therefore x = 2(3k) = 3(2k) and so x and x B. It follows then that x B by definition of intersection, thus C B. We conclude that B = C Proving Set Equivalences II n alternative prove uses membership tables where an entry is 1 if it a chosen (but fixed) element is in the set and 0 otherwise. (Exercise 13, p95): Show that B C = Ā B C

13 Proving Set Equivalences II Continued B C B C B C Ā B C Ā B C under a set indicates that an element is in the set. Since the columns are equivalent, we conclude that indeed, B C = Ā B C Generalized Unions & Intersections I In the previous example we showed that De Morgan s Law generalized to unions involving 3 sets. Indeed, for any finite number of sets, De Morgan s Laws hold. Moreover, we can generalize set operations in a straightforward manner to any finite number of sets. The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. n i = 1 2 n i=1 Generalized Unions & Intersections II Latex notation: \bigcup. The intersection of a collection of sets is the set that contains those elements that are members of every set in the collection. n i = 1 2 n i=1 Latex notation: \bigcap.

14 Computer Representation of Sets I There really aren t ways to represent infinite sets by a computer since a computer is has a finite amount of memory (unless of course, there is a finite representation). If we assume that the universal set U is finite, however, then we can easily and efficiently represent sets by bit vectors. Specifically, we force an ordering on the objects, say U = {a 1, a 2,..., a n } For a set U, a bit vector can be defined as { 0 if ai b i = 1 if a i for i = 1, 2,..., n. Computer Representation of Sets II Let U = {0, 1, 2, 3, 4, 5, 6, 7} and let = {0, 1, 6, 7} Then the bit vector representing is What s the empty set? What s U? Set operations become almost trivial when sets are represented by bit vectors. In particular, the bit-wise Or corresponds to the union operation. The bit-wise nd corresponds to the intersection operation. Computer Representation of Sets III Let U and be as before and let B = {0, 4, 5} Note that the bit vector for B is The union, B can be computed by = The intersection, B can be computed by What sets do these represent? = Note: If you want to represent arbitrarily sized sets, you can still do it with a computer how?

15 Conclusion Questions?

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

### 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

### 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

### 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

### 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:

### (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

### 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

### 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

### 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,

### SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

### Applications of Methods of Proof

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

### 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

### 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

### 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

### Math/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta

Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019 1

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### Sets and set operations: cont. Functions.

CS 441 Discrete Mathematics for CS Lecture 8 Sets and set operations: cont. Functions. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Set Definition: set is a (unordered) collection of objects.

### 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

### 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

### 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 (

### 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

### 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

### 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

### 1 / Basic Structures: Sets, Functions, Sequences, and Sums - definition of a set, and the use of the intuitive notion that any property whatever there

C H A P T E R Basic Structures: Sets, Functions, Sequences, and Sums.1 Sets. Set Operations.3 Functions.4 Sequences and Summations Much of discrete mathematics is devoted to the study of discrete structures,

### 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,

### 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

### Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour

Logic & Discrete Math in Software Engineering (CAS 701) Background Dr. Borzoo Bonakdarpour Department of Computing and Software McMaster University Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS

### Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1

Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms

### 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,

### Discrete Mathematics Set Operations

Discrete Mathematics 1-3. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.

### 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

### Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

### 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

### 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

### Discrete Mathematics. Some related courses. Assessed work. Motivation: functions. Motivation: sets. Exercise. Motivation: relations

Discrete Mathematics Philippa Gardner This course is based on previous lecture notes by Iain Phillips. K.H. Rosen. Discrete Mathematics and its Applications, McGraw Hill 1995. J.L. Gersting. Mathematical

### 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

### 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

### Notes on counting finite sets

Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

### 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,

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

### 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

### 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:

### CHAPTER 1. Basic Ideas

CHPTER 1 asic Ideas In the end, all mathematics can be boiled down to logic and set theory. ecause of this, any careful presentation of fundamental mathematical ideas is inevitably couched in the language

### 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

### axiomatic vs naïve set theory

ED40 Discrete Structures in Computer Science 1: Sets Jörn W. Janneck, Dept. of Computer Science, Lund University axiomatic vs naïve set theory s Zermelo-Fraenkel Set Theory w/choice (ZFC) extensionality

### In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)

Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### 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 3.3 Equivalence Relations

1 Section 3.3 Purpose of Section To introduce the concept of an equivalence relation and show how it subdivides or partitions a set into distinct categories. Introduction Classifying objects and placing

### 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 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

### 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

### 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

### Discrete Mathematics. Thomas Goller. July 2013

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

### Discrete Mathematics Lecture 5. Harper Langston New York University

Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = -1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### 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,

### 3.3 Proofs Involving Quantifiers

3.3 Proofs Involving Quantifiers 1. In exercise 6 of Section 2.2 you use logical equivalences to show that x(p (x) Q(x)) is equivalent to xp (x) xq(x). Now use the methods of this section to prove that

### 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

### Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since

Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the

### A Little Set Theory (Never Hurt Anybody)

A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### 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)

### A brief history of type theory (or, more honestly, implementing mathematical objects as sets)

A brief history of type theory (or, more honestly, implementing mathematical objects as sets) Randall Holmes, Boise State University November 8, 2012 [notes revised later; this is the day the talk was

### 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

Set (mathematics) From Wikipedia, the free encyclopedia A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental

### 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

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20

CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of

### 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.

### MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS

MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS ALLAN YASHINSKI Abstract. We explore the notion of well-definedness when defining functions whose domain is

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### COMPOSITES THAT REMAIN COMPOSITE AFTER CHANGING A DIGIT

COMPOSITES THAT REMAIN COMPOSITE AFTER CHANGING A DIGIT Michael Filaseta Mathematics Department University of South Carolina Columbia, SC 2208-0001 Mark Kozek Department of Mathematics Whittier College

### CHAPTER 2. Set, Whole Numbers, and Numeration

CHAPTER 2 Set, Whole Numbers, and Numeration 2.1. Sets as a Basis for Whole Numbers A set is a collection of objects, called the elements or members of the set. Three common ways to define sets: (1) A

### 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

### THE LANGUAGE OF SETS AND SET NOTATION

THE LNGGE OF SETS ND SET NOTTION Mathematics is often referred to as a language with its own vocabulary and rules of grammar; one of the basic building blocks of the language of mathematics is the language

### 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

### Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### 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

### 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

### 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,

### 2.3. Relations. Arrow diagrams. Venn diagrams and arrows can be used for representing

2.3. RELATIONS 32 2.3. Relations 2.3.1. Relations. Assume that we have a set of men M and a set of women W, some of whom are married. We want to express which men in M are married to which women in W.

### Basic Category Theory for Models of Syntax (Preliminary Version)

Basic Category Theory for Models of Syntax (Preliminary Version) R. L. Crole ( ) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. Abstract. These notes

### 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.

### Introduction to Relations

CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition: A binary relation from a set A to a set B is a subset R A B. If (a, b) R we say a is related

### 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

### Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### (Refer Slide Time 1.50)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Module -2 Lecture #11 Induction Today we shall consider proof

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### Notes 2 for Honors Probability and Statistics

Notes 2 for Honors Probability and Statistics Ernie Croot August 24, 2010 1 Examples of σ-algebras and Probability Measures So far, the only examples of σ-algebras we have seen are ones where the sample

### Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

### (Positive) Rational Numbers

(Positive) Rational Numbers Definition Definition 1 Consider all ordered pairs (, ) with, N Define the equivalance (, ) (y 1, ) = y 1 (1) Theorem 2 is indeed an equivalence relation, that is 1 (, ) (,

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES Contents 1. Probabilistic experiments 2. Sample space 3. Discrete probability

### RELATIONS AND FUNCTIONS

Chapter 1 RELATIONS AND FUNCTIONS There is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we

### Week 5: Quantifiers. Number Sets. Quantifier Symbols. Quantity has a quality all its own. attributed to Carl von Clausewitz

Week 5: Quantifiers Quantity has a quality all its own. attributed to Carl von Clausewitz Number Sets Many people would say that mathematics is the science of numbers. This is a common misconception among

### Binary Relations. Definition: A binary relation between two sets X and Y is a subset of X Y i.e., is a set of ordered pairs (x, y) X Y.

Binary Relations Definition: A binary relation between two sets X and Y is a subset of X Y i.e., is a set of ordered pairs (x, y) X Y. For a relation R X Y we often write xry instead of (x, y) R. We write

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.