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

Size: px
Start display at page:

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

Transcription

1 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 701) - McMaster University 1/26

2 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 2/26

3 Sets A set is a collection of objects called members or elements. We write α S to mean that α is a member of S (α S is the opposite). We write α 1,..., α n S to mean that α 1 S,..., and α n S. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 3/26

4 Sets Two sets are equal (i.e., S = T ) iff they have the same members: for every x, x S iff x T. S is said to be a subset of T (i.e., S T ) iff for every x, x S implies x T. Every set is a subset of itself. S = T iff S T and T S. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 4/26

5 Sets S is a proper subset of T (i.e., S T ), iff S T and S T. Sets are not ordered (e.g., {α, β} = {β, α}). Repetition in sets is not important (e.g., {α, α, β} = {α, β}). The empty set has no members. Hence, S for all S (why?). Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 5/26

6 Sets What are the concrete set that represent: {x x < 100 and x is prime} {x x = 0 or x = 1 or x = 2} Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 6/26

7 Sets We define S = {x x S} (complement) S T = {x x S or x T } (union) S T = {x x S and x T } (intersection) S T = {x x S and x T } (difference) Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 7/26

8 Sets We define S i = {x x S i for some i I} i I S i = {x x S i for all i I} i I Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 8/26

9 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 9/26

10 Relations The ordered pair of objects α and β is written as α, β. Then α, β = α 1, β 1 iff α = α 1 and β = β 1. Similarly, one can define and ordered n-tuple α 1,..., a n. One can also define a set of ordered pairs (e.g., { m, n m, n are natural numbers and m < n}). Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 10/26

11 Relations The Cartesian product of sets S 1,..., S n is defined by S 1 S n = { x 1,..., x n x 1 S 1,..., x n S n} Let S n = S S }{{} n An n-ary relation R on set S is a subset of S n. A special binary relation is the equality relation: or { x, y x, y S and x = y} { x, x x S} Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 11/26

12 Relations For a binary relation R, we often write xry to denote x, y R. R is reflexive on S, iff for any x S, xrx. R is symmetric on S, iff for any x, y S, whenever xry, then yrx. R is transitive on S, iff for any x, y, z S, whenever xry and yrz, then xrz. R is an equivalence relation iff R is reflexive, symmetric, and transitive. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 12/26

13 Relations Suppose R is an equivalence relation on S. For any x S the set x = {y S xry} is called the R-equivalence class of x. R-equivalence classes make a partition of S. The restriction of R to S 1 is the n-ary relation R S1. n Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 13/26

14 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 14/26

15 Functions A function (mapping) f is a set of ordered pairs such that if x, y f and x, z f, then y = z. The domain dom(f ) of f is the set {x x, y f for some y} The range ran(f ) of f is the set {y x, y f for some x} Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 15/26

16 Functions f (x) denotes the unique element in y ran(f ), where x dom(f ) and x, y f. If f is a function with dom(f ) = S and ran(f ) T, we say that f is a function from S to T and denote it by f : S T Similarly, one can define n-ary functions. Suppose f : S T is a function and S 1 S. The restriction of f to S 1 is the function f S 1 : S 1 T Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 16/26

17 Functions A function f : S T is onto if ran(f ) = T A function is one-to-one if f (x) = f (y) implies x = y. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 17/26

18 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 18/26

19 Set Cardinality Two sets S and T are equipotent (i.e., S T ) iff there is a one-to-one mapping from S onto T. is an equivalence relations (why?) A cardinal of a set S is denoted by S where: S = T iff S T. A set S is to be countably infinite, iff S = N. A set S is said to be countable, iff S N (i.e., S is finite or countably infinite). Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 19/26

20 Set Cardinality Theorem 1. A subset of a countable set is countable. Theorem 2. The union of any finite number of countable sets is countable. Theorem 3. The union of any countably many countable sets is countable. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 20/26

21 Set Cardinality Theorem 4. The Cartesian product of any finite number of countable sets is countable. Theorem 5. The set of all finite sequences with the members of a countable set as components is countable. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 21/26

22 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 22/26

23 Induction and Proofs Inductive definition of natural numbers: [1 ] 0 N. [2 ] For any n, if n N, then n N, where n is the successor of n. [3 ] n N only if n has been generated by [1] and [2]. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 23/26

24 Induction and Proofs Another inductive definition of natural numbers: [1 ] 0 S. [2 ] For any n, if n S, then n S Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 24/26

25 Induction and Proofs Theorem. Suppose R is a unary relation. If [1 ] R(0). [2 ] For any n N, if R(n), then R(n ). then R(n) for any n N. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 25/26

26 Example Prove that n = n(n + 1) 2 Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 26/26

SETS, RELATIONS, AND FUNCTIONS

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

More information

Math 3000 Running Glossary

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 (

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

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

More information

Sets, Relations and Functions

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

More information

Some Definitions about Sets

Some Definitions about Sets 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

More information

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 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

More information

Introduction to Relations

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

More information

Introducing Functions

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

More information

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

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

More information

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

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

More information

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

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

More information

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

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.

More information

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

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

More information

3. Equivalence Relations. Discussion

3. Equivalence Relations. Discussion 3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

More information

Discrete Mathematics, Chapter 5: Induction and Recursion

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

More information

Sets and set operations

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

More information

Studying Relationships

Studying Relationships Binary Relations Problem Problem Set Set Two Two checkpoint checkpoint due due in in the the box box up up front front if if you're you're using using a a late late period. period. Studying Relationships

More information

Sets and set operations: cont. Functions.

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.

More information

Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

More information

Lecture 17 : Equivalence and Order Relations DRAFT

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

More information

Mathematical Induction

Mathematical Induction Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

More information

Chapter 10. Abstract algebra

Chapter 10. Abstract algebra Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

More information

Week 5: Binary Relations

Week 5: Binary Relations 1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

More information

Sets and Cardinality Notes for C. F. Miller

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

More information

f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write

f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write Math 525 Chapter 1 Stuff If A and B are sets, then A B = {(x,y) x A, y B} denotes the product set. If S A B, then S is called a relation from A to B or a relation between A and B. If B = A, S A A is called

More information

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

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:

More information

INTRODUCTORY SET THEORY

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,

More information

Math 320 Course Notes. Chapter 7: Countable and Uncountable Sets

Math 320 Course Notes. Chapter 7: Countable and Uncountable Sets Math 320 Course Notes Magnhild Lien Chapter 7: Countable and Uncountable Sets Hilbert s Motel: Imagine a motel with in nitely many rooms numbered 1; 2; 3; 4 ; n; : One evening an "in nite" bus full with

More information

Lecture 16 : Relations and Functions DRAFT

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

More information

Section 3.3 Equivalence Relations

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

More information

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

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

More information

Sets and functions. {x R : x > 0}.

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.

More information

Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

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

More information

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

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

More information

Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

More information

POWER SETS AND RELATIONS

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

More information

CHAPTER 1. Basic Ideas

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

More information

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

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

More information

Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

More information

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS 1. Show that a nonempty collection A of subsets is an algebra iff 1) for all A, B A, A B A and 2) for all A A, A c A.

More information

2.1 Sets, power sets. Cartesian Products.

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

More information

Quiz 1. Ashish Carole Christos Eric George Jack Nick Tina

Quiz 1. Ashish Carole Christos Eric George Jack Nick Tina Massachusetts Institute of Technology 6.042J/18.062J, Spring 02: Mathematics for Computer Science March 1 Professor Albert Meyer and Dr. Radhika Nagpal revised April 28, 2002, 858 minutes Quiz 1 Circle

More information

The Language of Mathematics

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,

More information

Section 3.2 Partial Order

Section 3.2 Partial Order 1 Section 3.2 Purpose of Section To introduce a partial order relation (or ordering) on a set. We also see how order relations can be illustrated graphically by means of Hasse diagrams and directed graphs.

More information

Preference Relations and Choice Rules

Preference Relations and Choice Rules Preference Relations and Choice Rules Econ 2100, Fall 2015 Lecture 1, 31 August Outline 1 Logistics 2 Binary Relations 1 Definition 2 Properties 3 Upper and Lower Contour Sets 3 Preferences 4 Choice Correspondences

More information

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G. Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

More information

S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive. MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

More information

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

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

More information

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

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

More information

Computability Theory

Computability Theory CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.

More information

Where are we? Introduction to Logic 1

Where are we? Introduction to Logic 1 Introduction to Logic 1 1. Preliminaries 2. Propositional Logic 3. Tree method for PL 4. First-order Predicate Logic 5. Tree Method for FOL 6. Expressiveness of FOL Where are we? Introduction to Logic

More information

Methoδos Primers, Vol. 1

Methoδos Primers, Vol. 1 Methoδos Primers, Vol. 1 The aim of the Methoδos Primers series is to make available concise introductions to topics in Methodology, Evaluation, Psychometrics, Statistics, Data Analysis at an affordable

More information

Chapter 1. Sigma-Algebras. 1.1 Definition

Chapter 1. Sigma-Algebras. 1.1 Definition Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is

More information

Classical Analysis I

Classical Analysis I Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

More information

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

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

More information

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements. 3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but

More information

Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

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

More information

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

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

More information

Cartesian Products and Relations

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

More information

Geometry 2: Remedial topology

Geometry 2: Remedial topology Geometry 2: Remedial topology Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

More information

MEASURE ZERO SETS WITH NON-MEASURABLE SUM

MEASURE ZERO SETS WITH NON-MEASURABLE SUM INROADS Real Analysis Exchange Vol. 7(), 001/00, pp. 78 794 Krzysztof Ciesielski, Department of Mathematics, West Virginia University, Morgantown, WV 6506-610, USA. e-mail: K Cies@math.wvu.edu web page:

More information

Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set. Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

More information

Section 6.4 Closures of Relations

Section 6.4 Closures of Relations Section 6.4 Closures of Relations Definition: The closure of a relation R with respect to property P is the relation obtained by adding the minimum number of ordered pairs to R to obtain property P. In

More information

(Refer Slide Time: 1:41)

(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

More information

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 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

More information

Domain Theory: An Introduction

Domain Theory: An Introduction Domain Theory: An Introduction Robert Cartwright Rebecca Parsons Rice University This monograph is an unauthorized revision of Lectures On A Mathematical Theory of Computation by Dana Scott [3]. Scott

More information

arxiv: v1 [math.pr] 22 Aug 2008

arxiv: v1 [math.pr] 22 Aug 2008 arxiv:0808.3155v1 [math.pr] 22 Aug 2008 On independent sets in purely atomic probability spaces with geometric distribution Eugen J. Ionascu and Alin A. Stancu Department of Mathematics, Columbus State

More information

Relations and Functions

Relations and Functions Section 5. Relations and Functions 5.1. Cartesian Product 5.1.1. Definition: Ordered Pair Let A and B be sets and let a A and b B. An ordered pair ( a, b) is a pair of elements with the property that:

More information

In mathematics there are endless ways that two entities can be related

In mathematics there are endless ways that two entities can be related CHAPTER 11 Relations In mathematics there are endless ways that two entities can be related to each other. Consider the following mathematical statements. 5 < 10 5 5 6 = 30 5 5 80 7 > 4 x y 8 3 a b ( mod

More information

Math 421, Homework #5 Solutions

Math 421, Homework #5 Solutions Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition

More information

Lecture 4 -- Sets, Relations, Functions 1

Lecture 4 -- Sets, Relations, Functions 1 Lecture 4 Sets, Relations, Functions Pat Place Carnegie Mellon University Models of Software Systems 17-651 Fall 1999 Lecture 4 -- Sets, Relations, Functions 1 The Story So Far Formal Systems > Syntax»

More information

PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

More information

If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A.

If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A. Chapter 5 Cardinality of sets 51 1-1 Correspondences A 1-1 correspondence between sets A and B is another name for a function f : A B that is 1-1 and onto If f is a 1-1 correspondence between A and B,

More information

Sections 2.1, 2.2 and 2.4

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

More information

RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS Chapter 1 RELATIONS AND FUNCTIONS 1.1 Overview 1.1.1 Relation A relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product A B. The set of all first elements of the ordered

More information

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

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

More information

CS 3719 (Theory of Computation and Algorithms) Lecture 4

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

More information

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

More information

Notes 2 for Honors Probability and Statistics

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

More information

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. 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

More information

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

More information

Review for Final Exam

Review for Final Exam Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

More information

Arrow s Theorem on Fair Elections

Arrow s Theorem on Fair Elections Arrow s Theorem on Fair Elections JWR October 27, 2008 1 Introduction The fair way to decide an election between two candidates a and b is majority rule; if more than half the electorate prefer a to b,

More information

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

Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set. Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

More information

If we observe choices x at 4 and y at 1 then we have x y and y x. No one made choices like this.

If we observe choices x at 4 and y at 1 then we have x y and y x. No one made choices like this. 1 Experiment Did the subjects make choices as if they had a preference relation over bundles of (IC, HB)? If so, could we infer and predict future choices or offer advice about choices? In situation 2

More information

Chapter 1. Informal introdution to the axioms of ZF.

Chapter 1. Informal introdution to the axioms of ZF. Chapter 1. Informal introdution to the axioms of ZF. 1.1. Extension. Our conception of sets comes from set of objects that we know well such as N, Q and R, and subsets we can form from these determined

More information

MATHS 315 Mathematical Logic

MATHS 315 Mathematical Logic MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................

More information

Discrete Mathematics. Hans Cuypers. October 11, 2007

Discrete Mathematics. Hans Cuypers. October 11, 2007 Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

1.3 Induction and Other Proof Techniques

1.3 Induction and Other Proof Techniques 4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

More information

PRINCIPLE OF MATHEMATICAL INDUCTION

PRINCIPLE OF MATHEMATICAL INDUCTION Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION 4.1 Overview Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of

More information

We give a basic overview of the mathematical background required for this course.

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

More information

Automata and Formal Languages

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,

More information

Lecture 2 : Basics of Probability Theory

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

More information

Direct Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction

Direct Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following

More information

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

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.

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

Discrete Mathematics Set Operations

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.

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

More information

Math 317 HW #7 Solutions

Math 317 HW #7 Solutions Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

More information