CBMA2103 Discrete Maths cApr15 (rs)(M).pdf

CBMA2103
Discrete Mathematics
Faculty of Science and Technology
Copyright © Open University Malaysia (OUM)

CBMA2103
DISCRETE
MATHEMATICS
Assoc Prof Dr Abdullah Mohd Zin

First Edition, November 2008
Second Edition, April 2015 (rs)
Copyright © Open University Malaysia (OUM), April 2015, CBMA2103
All rights reserved. No part of this work may be reproduced in any form or by any means
without the written permission of the President, Open University Malaysia (OUM).
Project Directors: Prof Dato’ Dr Mansor Fadzil
Assoc Prof Dr Norlia T. Goolamally
Open University Malaysia
Module Writer: Assoc Prof Dr Abdullah Mohd Zin
Moderator: Prof Dr Mohammed Yusoff
Enhanced by: Dr Bahari Idrus
Universiti Kebangsaan Malaysia
Developed by: Centre for Instructional Design and Technology

Table of Contents
Course Guide xi - xv
Topic 1 Set 1
1.1 Concept of Set 2
1.1.1 Listing the Elements of Sets 2
1.1.2 Specifying Properties of Sets 3
1.1.3 Set Membership 4
1.1.4 Empty Set 5
1.1.5 Set of Numbers 6
1.2 Set Equality 7
1.3 Venn Diagram 8
1.4 Subset 10
1.5 Power Set 13
1.6 Set Operation 14
1.6.1 Union 15
1.6.2 Intersection 16
1.6.3 Disjoint Sets 18
1.6.4 Set Difference 19
1.6.5 Set Complementary 19
1.6.6 Characteristics of Set 20
1.7 Generalised Union and Intersection 22
1.8 Partition 23
1.9 Cartesian Product 24
Summary 26
Key Terms 27
References 27
Topic 2 Relation 28
2.1 Concept of Relation 29
2.2 Inverse Relation 32
2.3 Composition of Relations 34
2.4 Relations on a Set 36
2.4.1 Reflexive 37
2.4.2 Symmetric 37
2.4.3 Antisymmetric 38
2.4.4 Transitive 38
2.5 Digraph 39

 TABLE OF CONTENTS
iv
2.6 Partial Order 41
2.7 Equivalence Relation 43
Summary 44
Key Terms 45
References 46
Topic 3 Function 47
3.1 Concept of Function 48
3.2 Graph of a Function 50
3.3 Types of Function 52
3.3.1 Injective 52
3.3.2 Surjective 54
3.3.3 Bijective 55
3.4 Inverse of a Function 56
3.5 Functions Composition 58
3.6 Binary and Unary Operators 60
Summary 61
Key Terms 62
References 62
Topic 4 Sequence and Strings 63
4.1 Sequence 64
4.1.1 Types of Sequence 66
4.1.2 Subsequence 67
4.2 Sequence Operation 69
4.3 String 72
Summary 74
Key Terms 74
References 74
Topic 5 Propositional Logic 75
5.1 Proposition 76
5.1.1 Conjunction and Disjunction 77
5.1.2 Negation 78
5.2 Conditional Proposition 80
5.3 Biconditional Proposition 82
5.4 Tautologies, Contradictions and Logical Equivalence 84
5.5 Contrapositive and Converse 86
Summary 88
Key Terms 88
References 89

TABLE OF CONTENTS  v
Topic 6 Predicate Logic 90
6.1 Predicate Logic 91
6.2 Quantifiers 92
6.2.1 Universal Quantifier 93
6.2.2 Existential Quantifier 95
6.2.3 Combining Quantifiers 96
6.3 Generalised De Morgan Laws 98
6.4 Translating Sentences into Logical Expressions 99
Summary 100
Key Terms 101
References 101
Topic 7 Integer 102
7.1 Integer 102
7.1.1 Basic Operations 103
7.1.2 Order 104
7.1.3 Absolute Value 104
7.2 Mod 105
7.3 Divisor and Greatest Common Divisor 107
7.3.1 Divisors 107
7.3.2 Common Divisors 108
7.3.3 Greatest Common Divisors 109
7.3.4 Euclidean Algorithm 109
7.4 Prime Numbers 111
7.5 Cryptography 113
7.5.1 Private Key 114
7.5.2 Public Key 115
Summary 119
Key Terms 120
References 120
Topic 8 Counting 121
8.1 Basic Principle of Counting 121
8.1.1 Multiplication Principle 122
8.1.2 Addition Principle 123
8.1.3 Combining Principles 123
8.2 Permutation 127
8.3 Combination 129
8.4 Pigeonhole Principle 131
8.4.1 First Form 131
8.4.2 Second Form 132

 TABLE OF CONTENTS
vi
Summary 133
Key Terms 134
References 134
Topic 9 Matrices 135
9.1 Matrices 135
9.1.1 Equal Matrices 137
9.1.2 Matrix Addition 137
9.1.3 Matrix Multiplication 138
9.1.4 Identity Matrix 139
9.1.5 Power of Square Matrices 139
9.1.6 Matrix Transpose 140
9.1.7 Zero-One Matrices 140
9.2 Matrices of Relation 141
9.2.1 Representing Relations as Matrices 141
9.2.2 Using Matrices for Analysis of Relations 143
9.2.3 Checking for Transitivity 145
Summary 148
Key Terms 149
References 149
Topic 10 Introduction to Graphs 151
10.1 The Concept of Graphs 151
10.2 Types of Graphs 154
10.2.1 Directed Graphs 154
10.2.2 Simple Graphs 156
10.2.3 Weighted Graphs 157
10.2.4 Complete Graphs 157
10.2.5 Cycles 158
10.2.6 n-cube 159
10.2.7 Bipartite Graphs 159
10.2.8 Complete Bipartite Graphs 161
10.3 Subgraphs 164
Summary 165
Key Terms 166
References 166

TABLE OF CONTENTS  vii
Topic 11 Path and Cycle 167
11.1 Path 168
11.2 Connected Graph 171
11.3 Components 172
11.4 Euler Path and Cycle 174
11.5 Hamilton Path and Cycle 179
Summary 181
Key Terms 182
References 182
Topic 12 Graph Representation and Isomorphism 183
12.1 Graph Representation 184
12.1.1 Adjacency Matrix 184
12.1.2 Incidence Matrix 187
12.2 Isomorphism 190
Summary 195
Key Terms 195
References 195
Topic 13 Planar Graph 197
13.1 Planar Graph 198
13.2 Graph Colouring 201
Summary 204
Key Terms 205
References 205
Topic 14 Tree 207
14.1 Concept of Trees 208
14.2 Important Terminology 210
14.3 Binary Tree 213
14.4 Tree Isomorphism 218
14.4.1 Basic Concept of Isomorphism 219
14.4.2 Rooted Isomorphism 221
14.4.3 Binary Isomorphism of Trees 222
Summary 226
Key Terms 226
References 227

X COURSE ASSIGNMENT GUIDE
xxvi

COURSEGUIDE

COURSE GUIDE DESCRIPTION
You must read this Course Guide carefully from the beginning to the end. It tells
you briefly what the course is about and how you can work your way through
the course material. It also suggests the amount of time you are likely to spend in
order to complete the course successfully. Please keep on referring to Course
Guide as you go through the course material as it will help you to clarify
important study components or points that you might miss or overlook.
INTRODUCTION
CBMA2103 Discrete Mathematics is one of the courses offered by Faculty of
Science and Technology at Open University Malaysia (OUM). This course is
worth 3 credit hours and should be covered over 8 to 15 weeks.
COURSE AUDIENCE
This course is offered to learners undertaking the Bachelor of Information
Technology programme. This module aims to impart the importance of Discrete
Mathematics in digital electronics as well as Information Technology. This
module is basically the introductory course which focuses on concepts and
techniques of Discrete Mathematics.
As an open and distance learner, you should be acquainted with learning
independently and being able to optimise the learning modes and environment
available to you. Before you begin this course, please confirm the course material,
the course requirements and how the course is conducted.
STUDY SCHEDULE
It is a standard OUM practice that learners accumulate 40 study hours for every
credit As such, for a three-credit hour course, you are expected to spend 120
study hours. Table 1 gives an estimation of how the 120 study hours could be
accumulated.

 COURSE GUIDE
xii
Table 1: Estimation of Time Accumulation of Study Hours
Study Activities
Study
Hours
Briefly go through the course content and participate in initial discussion 3
Study the module 60
Attend 3 to 5 tutorial sessions 10
Online participation 12
Revision 15
Assignment(s), Test(s) and Examination(s) 20
TOTAL STUDY HOURS 120
COURSE OUTCOMES
By the end of this course, you should be able to:
1. Write statements using mathematical language;
2. Develop mathematical arguments using language;
3. Discuss the concept of integers and its role in modelling;
4. Summarise the concept of graphs and trees; and
5. Apply the concept of graph and tree models to solve problems.
COURSE SYNOPSIS
This course is divided into 14 topics. The synopsis for each topic can be listed as
follows:
Topic 1 introduces the concept of sets and subsets. It also shows the theory
application of sets operations such as intersection, union, difference and
products.
Topic 2 discusses the concept of relation. This topic introduces appropriate
methods for representing relations between objects. It also discusses some of the
properties of relations.

COURSE GUIDE  xiii
Topic 3 elaborates on the concept of function which is basically a special kind of
relation that has been discussed in Topic 2. This topic also explains the graphical
representations of functions and the different types of functions.
Topic 4 discusses the concept of sequence and strings. It also shows the
operations on sequence and strings.
Topic 5 explains the concept of propositions logic. It also explains the type of
logic that deals with propositions and how to reason on propositions.
Topic 6 elaborates on the concept of predicate logic. This topic also teaches you
how to write a statement using the predicate logic concept.
Topic 7 discusses the applications of integers. It also explains the use of integers
in cryptography.
Topic 8 introduces several tools for counting. It also briefly discusses the concept
of recurrence relation.
Topic 9 explains the concept of matrices and applies all matrices operations. This
topic also shows how to use matrices to model relationships between two sets.
Topic 10 introduces the basic concept of graphs and subgraphs. By the end of this
topic, you should be able to identify eight different types of graphs.
Topic 11 discusses the concept of path and cycle in a graph. It also introduces
applications of graphs by using Euler and Hamilton path and cycle.
Topic 12 shows the representation of graphs using matrices. It also explains the
concept of isomorphism.
Topic 13 discusses the concept of planar graphs in problem solving. This topic
also shows how to solve the map colouring problem.
Topic 14 discusses the concept of trees and the characteristics of different types of
trees. Then, you will be introduced to the application of the tree isomorphism
concept.

 COURSE GUIDE
xiv
TEXT ARRANGEMENT GUIDE
Before you go through this module, it is important that you note the text
arrangement. Understanding the text arrangement will help you to organise your
study of this course in a more objective and effective way. Generally, the text
arrangement for each topic is as follows:
Learning Outcomes: This section refers to what you should achieve after you
have completely covered a topic. As you go through each topic, you should
frequently refer to these learning outcomes. By doing this, you can continuously
gauge your understanding of the topic.
Self-Check: This component of the module is inserted at strategic locations
throughout the module. It may be inserted after one sub-section or a few sub-
sections. It usually comes in the form of a question. When you come across this
component, try to reflect on what you have already learnt thus far. By attempting
to answer the question, you should be able to gauge how well you have
understood the sub-section(s). Most of the time, the answers to the questions can
be found directly from the module itself.
Activity: Like Self-Check, the Activity component is also placed at various
locations or junctures throughout the module. This component may require you
to solve questions, explore short case studies, or conduct an observation or
research. It may even require you to evaluate a given scenario. When you come
across an Activity, you should try to reflect on what you have gathered from the
module and apply it to real situations. You should, at the same time, engage
yourself in higher order thinking where you might be required to analyse,
synthesise and evaluate instead of only having to recall and define.
Summary: You will find this component at the end of each topic. This component
helps you to recap the whole topic. By going through the summary, you should
be able to gauge your knowledge retention level. Should you find points in the
summary that you do not fully understand, it would be a good idea for you to
revisit the details in the module.
Key Terms: This component can be found at the end of each topic. You should go
through this component to remind yourself of important terms or jargon used
throughout the module. Should you find terms here that you are not able to
explain, you should look for the terms in the module.
References: The References section is where a list of relevant and useful
textbooks, journals, articles, electronic contents or sources can be found. The list
can appear in a few locations such as in the Course Guide (at the References

COURSE GUIDE  xv
section), at the end of every topic or at the back of the module. You are
encouraged to read or refer to the suggested sources to obtain the additional
information needed and to enhance your overall understanding of the course.
PRIOR KNOWLEDGE
No prior knowledge required.
ASSESSMENT METHOD
Please refer to the myINSPIRE.
TAN SRI DR ABDULLAH SANUSI (TSDAS)
DIGITAL LIBRARY
The TSDAS Digital Library has a wide range of print and online resources for the
use of its learners. This comprehensive digital library provides access to more
than 30 online databases comprising e-journals, e-theses, e-books and more.
Examples of databases available are EBSCOhost, ProQuest, SpringerLink,
Books24x7, InfoSci Books, Emerald Management Plus and Ebrary Electronic
Books. As an OUM learner, you are encouraged to make full use of the resources
available through this library.

 INTRODUCTION
Did you know that the concept of set is fundamental to mathematics and
computer science? This is because everything mathematical starts with sets. For
example:
(a) Relationships between two objects are represented as a set of ordered pairs
of objects (the concept of ordered pair is defined using sets);
(b) Natural numbers (the basis of other numbers) are also defined using sets;
(c) The concept of function (a special type of relation) is based on sets; and
(d) Graphs and digraphs consisting of lines and points are described as an
ordered pair of sets.
So what does a set mean? A set is an unordered collection of objects, and, as such,
a set is determined by the objects it contains. Then we have set theory which is an
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 Sets
By the end of the topic, you should be able to:
1. Describe the concept of sets;
2. Define set equality;
3. Illustrate sets by using Venn diagrams;
4. Discuss subset and power set;
5. Apply the operations on sets;
6. Solve generalised union and intersection; and
7. Summarise the concept of partition and Cartesian products.
LEARNING OUTCOMES

 TOPIC 1 SETS
2
important language and tool for reasoning. It is a basis for mathematics. Set
theory is important for computer science because it is a useful tool for
formalising and reasoning about computation and the objects of computation.
CONCEPT OF SET
Let us start the lesson by learning the concept of set; firstly, its definition.
For example:
(a) The collection of all learners taking this course;
(b) The collection of all lecturers at the university; and
(c) The collection of odd numbers between one to fifteen.
1.1.1 Listing the Elements of Sets
Generally, there are several ways to describe sets. One way of describing a set
that has a finite number of elements is by listing the elements of the set between
curly brackets.
Notation
A = {elements of set A}
where A is the name of the set. It could be any other variable name. Other
examples are C, D, E and so on. Normally the name of the set is denoted by
capital letters while elements of a set are with small letters, for example P = {x, y,
z}. Let us look at other examples.
Example 1.1a
Given set A,
A = {1, 2, 3, 4}
describes a set A made up of the four elements 1, 2, 3, and 4.
1.1
Definition 1.1a: A set is any well-defined collection of objects, called
elements or members of the set.

TOPIC 1 SETS  3
A set is determined by its elements, and the order in which the elements of a set
are listed does not matter. Another way in writing the same set A is as follows:
A = {1, 3, 4, 2}
The elements comprised in a set could be listed more than once as they are
assumed to be distinct. For this reason, we may also describe the same set A as:
A = {1, 2, 2, 3, 4}
Listing the elements of set could be in order or unordered or distinct.
To summarise, there are three ways for us to represent a set as simplified in
Table 1.1.
Table 1.1: Representation of Elements in a Set
Type of Elements
in a Set
Representation Description
Order A = {1, 2, 3, 4} A set is determined by its elements and order in
which the elements might be listed.
Unorder A = {1, 3, 4, 2} A set is determined by its elements and not by
any particular order in which the elements
might be listed.
Distinct
(unequal)
A = {1, 2, 2, 3, 4} The elements making up a set are assumed to
be distinct, and although for some reason
we may duplicate them out in a list, only
one occurrence of each element is in the set.
Example 1.1b
The set V of all vowels in the English alphabet can be written as
V = {a, e, i, o, u}
1.1.2 Specifying Properties of Sets
Sometimes it is inconvenient or impossible to describe a set by listing all of its
elements. Another useful way to define a set is by specifying a property
that the elements of the set have in common. If a set is a large finite set or an
infinite set, we can describe it by listing a property necessary for membership.
Let us check out the following example.

 TOPIC 1 SETS
4
Example 1.1c
The set
B = {x | x is a positive and even integer}
describes the set B made up of all positive and even integers; that is, B
consists of the integer 2, 4, 6, 8 and so on.
The vertical bar "|" is read as "such that". Set B can be read as "B equals the set of
all x such that x is a positive and even integer."
1.1.3 Set Membership
Given a description of a set X such as
A = {1, 2, 3, 4} or B = {x| x is a positive and even integer}
with element x, we can determine whether or not x belongs to X.
If x is a member in set X, we write it as x X; otherwise, we write it as x  X.
If X is a finite set, the number of elements in X, is denoted as |X| called
cardinality.
These relationships can be simplified in Table 1.2.
Table 1.2: Membership Representation
Symbols Description
 Is a member/element
 Is not a member/element
Let us look at the following examples of set membership.
Explain how to read the set notation. Give an example.
SELF-CHECK 1.1

TOPIC 1 SETS  5
Example 1.1d
Let A = {1, 3, 5, 7}. Then 1  A, 3  A, but 2 A.
Therefore |A| = 4.
Example 1.1e
A is a set consisting of the first five positive integers: A = {1, 2, 3, 4, 5}.
So 2  A but 6  A.
Therefore |A| = 5.
Example 1.1f
Let N = {1, 2, 3, ⁄} be a set of positive integers,
(a) A = {1, 4, 9,⁄., 64, 81}
= {x2 | x  N and x2 < 100}
(b) B = {1, 4, 9, 16}
= {y2 | y  N and y2 16}
(c) C = {2, 4, 6, 8, ⁄⁄⁄}
= {2k | k  N }
A and B are called finite sets while C is called an infinite set.
1.1.4 Empty Set
Did you know that there is a special set that has no elements? This set is called
the empty set or null set, and is denoted by  or { }. This empty set can be
simplified by using symbols, as shown in Figure 1.1.
Figure 1.1: Representation of an empty set

 TOPIC 1 SETS
6
1.1.5 Set of Numbers
The following Table 1.3 shows you the sets that are normally used when dealing
with numbers.
Table 1.3: Set of Numbers
Set of Numbers Notation Description
Integers Z Z = {x | x is an integer}
Example: ⁄-3,-2,-1,0,1,2,3,⁄
Positive integers Z + Z + = {x | x is a positive integer}
Example: 1,2,3⁄
Natural numbers V N = {x | x is a positive integer}
Example: 1,2,3⁄
Real numbers R R = {x | x is a real number}
ACTIVITY 1.1
1. Let A= {1, 2, 4, a, b, c}. Identify each of the following as true or false:
(a) 2  A
(b) 3  A
(c) c  A
(d)   A
(e) { }  A
(f) A  A
2. Let A= {x | x is an integer and x < 6}. Identify each of the following as
true or false.
(a) 3  A
(b) 6  A
(c) 5  A
(d) 8  A
(e) -8  A
(f) -4  A

TOPIC 1 SETS  7
Visit this website which provides detail information on the set theory. It will be
helpful for those who are still confused with sets and is worth reading. Enjoy!
http://en.wikipedia.org/wiki/Naive_set_theory
SET EQUALITY
A set is completely known when its members are all known. Thus, we say two
sets of A and B are equal if they have the same elements and we write A = B. Let
us look at the following examples to understand more on this matter.
Example 1.2a
If A = {1, 2, 3} and B = {x|x is a positive integer and x2 < 12}, then A = B.
Example 1.2b
If A = {BASIC, PASCAL, ADA}, B = {ADA, BASIC, PASCAL} and C = { ADA,
ADA, BASIC, PASCAL, BASIC} then A = B = C.
1.2
3. Describe the following sets by listing their elements.
(a) The set of all positive integers that are less than 10.
(b) {x | x  Zand x2 < 12}
4 Write the following sets in the form {x | p (x)}, where p (x) is a
property that describes the elements of the sets.
(a) {2, 4, 6, 8, 10}
(b) {a, e, i, o, u}
(c) {1, 8, 27, 64, 125}
(d) {-2, -1, 0, 1, 2}
5. What is the cardinality of
(a) empty set?
(b) {}?
(c) {1, 2, 3, 4, 7}?
(d) {a, b, b, c, d, d}?

 TOPIC 1 SETS
8
Example 1.2c
If A = {x | x2 + x – 6 = 0},
B = {2, –3}
A = B since x2 + x – 6 = 0 can be factorised into (x – 2) (x + 3) = 0, giving x = 2 and
x = – 3.
VENN DIAGRAM
What is the purpose of Venn diagrams? The purpose of Venn diagrams is to
provide pictorial views of a set. Historically, the idea of Venn diagrams was first
proposed by a mathematician by the name of John Venn.
1.3
1. Let A = {1, 2, 3, 4, 5}. Which of the following sets are equal to A?
(a) {4, 1, 2, 3, 5}
(b) {2, 3, 4}
(c) {1, 2, 3, 4, 5, 6}
(d) {x | x is an integer and x2  25}
(e) {x | x is a positive integer and x  5}
(f) {x | x is a positive rational number and x  5}
2. Which of the following sets are empty sets?
(a) {x|x is a real number and x2 – 1 = 0}
(b) {x|x is a real number and x2 + 1 = 0}
(c) {x|x is a real number and x2 = –9}
(d) {x|x is a real number and x = 2x + 1}
(e) {x|x is a real number and x = x + 1}
3. Determine whether each of the following pairs of sets are equal?
(a) {1, 3, 3, 3, 5, 5, 5, 5, 5}, {5, 3, 1}
(b) {{1}}, {1, {1}}
(c) , {}
ACTIVITY 1.2

TOPIC 1 SETS  9
In Venn diagrams, the universal set E will normally be denoted by a rectangle,
while sets within E will be denoted by circles as shown in Figure 1.2.
Figure 1.2: Venn diagram
As for subsets of the universal set, they are drawn as circles. The figures inside of
a circle represent the elements of the set (Figure 1.3).
Figure 1.3: Subset representation
How do we represent three sets? We represent three sets by using three
overlapping circles, as shown in Figure 1.4.
Figure 1.4: Three sets representation

 TOPIC 1 SETS
10
SUBSET
We have learned a little on subset in the previous subtopic. Now let us learn
more on subsets; firstly, its definition.
The Venn diagram in Figure 1.5 represents the subset for sets A and B.
Figure 1.5: Subset for sets A and B
Let us check out some examples on subsets.
Example 1.4a
If C = {1, 3} and A = {1, 2, 3, 4}
Then C is a subset of A, or C A
We can present it in a Venn diagram as shown in Figure 1.6.
1.4
Definition 1.4a: If every element of A is also an element of B, that is
if whenever x  A then x  B, we say that A is a subset of B or A is
contained in B, and we write A  B.

TOPIC 1 SETS  11
Figure 1.6: C subset of A
Example 1.4b
Let
A = {1, 2, 3, 4, 5, 6};
B = {2, 4, 5}; and
C = {1, 2, 3, 4, 5}.
B  A, B  C and C  A
Example 1.4c
Let
A = {1, 2, 3, 4, 5, 6}
B = {3, 7}
B is not a subset of A, so we write B A.
Figure 1.7: Venn diagram for Example 1.4c

 TOPIC 1 SETS
12
Any set X is a subset of itself, since any element in X is in X, so X  X.
As for the empty set, it is a subset of any set. So for any set X,   X.
We can define set equality by using the concept of subsets as follows:
Definition 1.4b: If X is a subset of any set Y and X is not equal to Y, we
say that X is a proper subset of Y and we write X Y.
Definition 1.4c: Two sets, A and B are equal and can be written as A = B if
A B and B A.
ACTIVITY 1.3
1. Let A = {1, 2, 5, 8, 11}. Identify each of the following statements
as true or false.
(a) {5, 1}  A
(b) {8, 1}  A
(c) {1, 8, 2, 11, 5}  A
(d)   A
(e) {1, 6}  A
(f) {2}  A
(g) {3}  A
(h) A  {11, 2, 5, 1, 8, 4}
2. Suppose that A = {2, 4, 6}, B= {2, 6}, C = {4, 6} and D = {4, 6, 8}.
Determine whether
(a) A  B
(b) B  C
(c) C  D

TOPIC 1 SETS  13
POWER SET
What is a power set? The following describes the meaning of a power set.
Let us look at some examples of power sets.
Example 1.5a
If A = {a, b, c }, the members of  (A) are
 (A) = {, {a}, {b}, {c }, {a, b}, {a, c }, {b, c }, {a, b, c }}
All but {a, b, c } are proper subsets of A. You may also write (A) as ({a, b, c})
The number of elements in a power set is given by the following theorem.
Example 1.5b
For set A in Example 1.5a,
|A| = 3, then |P (A) | = 23 = 8
Example 1.5c
P () = , || = 0 then|P ()|= 20 = 1
1.5
3. Use a Venn diagram to illustrate the relationship A  B and B  C.
4. Suppose that A, B and C are sets, such that A  B and B  C. Show
that A  C.
Definition 1.5a: If A is a set, then the set of all subsets of A including the
empty set and itself is called the power set of A and is denoted by  (A).
Theorem 1.5a: If X  = n then P (X ) = 2n

 TOPIC 1 SETS
14
Example 1.5d
The power set of the empty set has two subsets,  and {}.
2 = 2
P ({}) = {
SET OPERATION
This subtopic discusses several operations that will combine given sets to
produce new sets. These operations are union, intersection, disjoint sets, set
difference and set complementary.
1.6
1. Find the power set of each of the following sets.
(a) {a}
(b) {a, b}
(c) {, {}}
2. Can you conclude that A = B, if A and B are two sets with the same
power set?
3. How many elements does each of the following sets have?
(a) P ({a, b, {a, b}})
(b) P({, {a, {a}, {{a}}})
(c) P(P())
4. Determine whether each of the following sets is the power set of a set.
(a) 
(b) {, {a}}
(c) {, {a}, {, a}}
(d) {, {a}, {b}, {a, b}}
ACTIVITY 1.4

TOPIC 1 SETS  15
1.6.1 Union
What does union mean?
The union of two sets can be illustrated by using a Venn diagram (Figure 1.8).
Figure 1.8: Union of set A and set B
Let us look at an example.
Example 1.6a
Let A = {a, b, c, d, e} and B = {b, d, r, s}.
A B consists of all the elements that belong to either A or B, so
A B = {a, b, c, d, e} {b, d, r, s} = {a, b, c, d, e, r, s}.
Definition 1.6a: If A and B are sets, their union can be defined as a
set consisting of all elements that belong to A or B, and is denoted by A 
B. Thus A  B = {x | x  A or x  B}.

 TOPIC 1 SETS
16
Figure 1.9: A  B
1.6.2 Intersection
What does it mean by intersection? Let us look its definition.
The intersection of the two sets A and B can be illustrated as follows (Figure 1.10).
Figure 1.10: Intersection of set A and set B
Definition 1.6b: If A and B are sets, their intersection can be defined as a set
consisting of all elements that belong to both A and B. The intersection of A
and B is denoted by A  B. Thus A  B = {x | x  A and x  B}.

TOPIC 1 SETS  17
Let us check out the following example.
Example 1.6b
Let
A = {a, b, c, d, e, f}
B = {b, e, f, r, s}
C = {a, t, u, v}.
A  B = {b, e, f } since elements b, e, and f belong to both A and B.
Figure 1.11: A  B
Similarly, A  C = {a}.
Figure 1.12: A  C

 TOPIC 1 SETS
18
B  C = {}, since there are no elements that belong to both B and C.
Figure 1.13: B  C
1.6.3 Disjoint Sets
What do disjoint sets mean?
We can illustrate the disjoint of two sets with a Venn diagram as follows
(Figure 1.14).
Figure 1.14: Disjoint of set A and set B
Definition 1.6c: Two sets that have no common elements are called disjoint
sets. A  B = .

TOPIC 1 SETS  19
1.6.4 Set Difference
Now let us look at set difference; firstly, its meaning.
Figure 1.15: A – B
1.6.5 Set Complementary
What does set complementary mean?
Figure 1.16: Set complementary representation
Definition 1.6d: If A and B are sets, their difference can be defined as a
set consisting of all elements in A that are not in B. The difference between
A and B is denoted by A – B. Thus A – B = {x | x  A and x  B}.
Definition 1.6e: Let E be a universal set and let A be a subset of E. The set of E – A
consisting of all elements of E that are not elements of A is called the complement
of A and is denoted byA .

 TOPIC 1 SETS
20
Let us consider these examples.
Example 1.6c
Let E = {1, 2, 3, 4, 5, 6} and A = {1, 2}.
Therefore,
A = E – A
= {1, 2, 3, 4, 5, 6} – {1, 2}
= {3, 4, 5, 6}.
Figure 1.17: Ā = E – A
1.6.6 Characteristics of Set
Lastly, let us look at characteristics of sets.
Let us look at Table 1.4 which summarises nine theorems of sets.
Table 1.4: Theorem of Sets
Laws
(a) Associative Laws (A  B)  C = A  (B  C)
(A  B)  C = A  (B  C)
(b) Commutative Laws A  B = B  A
A  B = B  A
Theorem 1.6a: Let E be a universal set and let A, B and C be subsets of E. The
following properties hold (see Table 1.4).

TOPIC 1 SETS  21
(c) Distributive Laws A  (B  C) = (A  B)  (A  C)
A  (B  C) = (A  B)  (A  C)
(d) Identity Laws A   = A
A  A = A
(e) Complement Laws A  Ā = E
A  Ā = E
(f) Idempotent Laws A  A = A
A  A = A
(g) Bound Laws A  E = E
A   = 
(h) Absorption Laws A  (A  B) = A
A  (A  B) = A
(i) De Morgan's Laws for Sets
 
   
  
 
   
  
ACTIVITY 1.5
In each statement below, write „true‰ if the statement is true; otherwise,
give a counter example. The sets X, Y and Z are subsets of a universal
set E.
(a) X  (Y – Z ) = (X Y ) – (X Z ) for all sets X, Y and Z.
(b) (X – Y )  (Y – X ) = for all sets X and Y.
(c) X – (Y Z ) = (X – Y ) Z for all sets X, Y and Z.
(d) – –

X Y Y X for all sets X and Y.
(e) 
X Y  X for all sets X and Y.
(f) (X Y)  (Y – X ) = X for all sets X and Y.

 TOPIC 1 SETS
22
GENERALISED UNION AND INTERSECTION
Firstly, let us get to know the definition of generalised union.
If S = {A1
, A2
, ⁄ An
}, we can write U S =
1
n
i
i
A

 , i = 1, 2, ...n
Next is the definition of generalised intersection.
If S = { A1
, A2
, ⁄, An
}, we can write I S =
1
n
i
i
A

 , i = 1, 2, ...n
Let us check Example 1.7a.
Example 1.7a
Suppose A1 = {1, 2}, A2 = {3, 4} and A3 = {5, 6} then
S = {A1, A2, A3}
= { {1, 2} , {3, 4} , {5, 6} }
Therefore,
U S =
1
n
i
i
A

 = {1, 2, 3, 4, 5, 6 }
 S =
1
n
i
i
A

 = { }
1.7
Definition 1.7a: The generalised union of an arbitrary family, S, of sets are
those elements x belonging to at least one set X in S. Formally,
U S = {x | x  X for some X  S}.
Definition 1.7b: The generalised intersection of an arbitrary family, S, of sets
are those elements x belonging to all sets X in S. Formally,
S = {x | x  X for all X  S}

TOPIC 1 SETS  23
PARTITION
A partition of a set X divides n into non-overlapping subsets. More formally,
Let us see Example 1.8a.
Example 1.8a
Since each element of
X = {1, 2, 3, 4, 5, 6, 7, 8}
is in exactly one member of
S = {{1, 4, 5}, {2, 6}, {3}, {7, 8}}
S is a partition of X
Notice that if S is a partition of X, S is a pairwise disjoint, and U S = X.
1.8
Definition 1.8a: Let S be a collection of non-empty subsets of set X. S is
said to be a partition of X, if every element of X belongs to exactly one
member of S.
Let Ai be a set of integers from 1 to 4. Assume A1 = {a, b, c }, A2 = {b, c,
d }, A3 = {b, c, f } and A4 = {b, c, e, f, g }. Find
(a)
1
n
i
i
A


(b)
1
n
i
i
A


ACTIVITY 1.6

 TOPIC 1 SETS
24
CARTESIAN PRODUCT
What can we say about Cartesian products?
An ordered pair (a, b), is considered distinct from the ordered pair (b,
a), unless, of course a = b. To put it another way, (a, b) = (c, d) if and only if
a = c and b = d.
Let us look at Example 1.9a.
Example 1.9a
If X = {1, 2, 3} and Y = {a, b}, then
X × Y = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}
Y × X = {(a, 1), (b, 1), (a, 2), (b, 2), (a, 3), (b, 3)}
X × X = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
Y × Y = {(a, a), (a, b), (b, a), (b, b)}
The example above shows that, in general, X × Y  Y × X.
1.9
Definition 1.9a: If A and B are sets, we let A  B denote the set of all ordered
pairs (a, b) where a  A and b  B. We call A  B the Cartesian product of A
and B.
ACTIVITY 1.7
Determine whether each set D below is a partition of set Y
Y = {x | x is integer and 1  x  5}
(a) D = {{1}, {2, 3}, {4}, {5}}
(b) D = {{1, 4}, {2}, {3,5}}
(c) D = {{1}, {4}, {2}, {3,5}}
(d) D = {{1}, {4}, {2}, {3}, {5}}
(e) D = {{1, 3}, {2}, {4, 5}}
(f) D = {{7, 4}, {2, 6}, {3, 5}}

TOPIC 1 SETS  25
Let us look at an example on this.
Example 1.9b
If X = {1, 2} Y = {a, b} Z = {, }, then
X × Y × Z = {(1, a, ), (1, a, ), (1, b, ), (1, b, ), (2, a, ), (2, a, ), (2, b, ), (2, b, )}
The following are some basic properties of Cartesian products:
A ×  = 
A × (B  C) = (A × B)  (A × C)
(A  B) × C = (A × C)  (B × C)
| A × B | = | B × A | = |A|× |B | A and B is finite set.
Definition 1.9b: The Cartesian product of sets X1, X2, ⁄ Xn is defined to be the
set of all n-tuples (x1, x2,⁄. xn) where xi  Xi for i = 1, ⁄ n; it is denoted by X1 ×
X2 ×⁄× Xn.
Theorem 1.9a: | X × Y | = | X | . | Y | and | X × Y × Z | = | X | . | Y | . | Z |. In
general, we have | X1 × X2 × ⁄. X × Xn | = | X1 | . | X1 | . | X2 | ……. | Xn |
1. Find x or y so that the following statements are true.
(a) (x, 3) = (4, 3)
(b) (a, 3y) = (a, 9)
(c) (3x + 1, 2) = (7, 2)
(d) (C++, PASCAL) = (x, y)
ACTIVITY 1.8

 TOPIC 1 SETS
26
 A set is any well-defined collection of objects, called elements or members of
the set.
 A set is completely known when its members are all known. Thus, we say two
sets of A and B are equal if they have the same elements, and we write A = B.
 A Venn diagram illustrates the universal set E as a rectangle, while sets
within E will be denoted by circles.
 Set A is a subset of set B if every element of A is also an element of B, A
B.
 If A is a set, then the set of all subsets of A including the empty set and
itself is called the power set of A, and is denoted by  (A).
 The operations of sets include union, intersection and difference.
 The generalised union of an arbitrary family, S, of sets are those elements x
belonging to at least one set X in S. Formally, U S = {x | x  X for some X  S}
 The generalised intersection of an arbitrary family, S, of sets are those
elements x belonging to all set X in S. Formally, S = {x | x  X for all X  S}
2. In each of the statements below, write „true„ if the statement is
true; otherwise, give a counter example. The set X, Y and Z are
subsets of a universal set E. Assume that the universe for Cartesian
products is E × E.
(a)   
X Y X Y for all sets X and Y.
(b) X × (Y  Z) = (X × Y)  (X × Z) for all sets X, Y and Z.
(c) X × (Y – Z) = (X × Y) – (X × Z) for all sets X, Y and Z.
(d) X – (Y × Z) = (X – Y) × (X – Z ) for all sets X, Y and Z.
(e) X  (Y × Z) = (X  Y) × (X  Z) for all sets X, Y and Z.

TOPIC 1 SETS  27
 A partition of a set X divides set X into n non-overlapping subsets.
 If A and B are sets, we let A  B denote the set of all ordered pairs (a, b)
where a  A and b  B. We call A  B the Cartesian product of A and B.
Cartesian product
Disjoint set
Elements
Empty set
Generalised intersection
Generalised union
Intersection
Partition
Power set
Set complement
Set difference
Set equality
Set membership
Sets
Subsets
Union
Venn diagram
Glosser, G. (2015). Introductions to Sets. Retrieved from http://www.math
goodies.com/lessons/sets/
Johnsonbaugh, R. (2014). Discrete mathematics (7th ed.). Edinburgh Gate:
Pearson.
Lipschutz, S., & Lipson, M. (2009). SchaumÊs outline of discrete mathematics
(revised 3rd ed.). New York: McGraw Hill.
Rosen, K. (2011). Discrete mathematics and its application (7th ed.). New York:
McGraw Hill.
Set (mathematics). (2015). Retrieved from http://en.wikipedia.org/wiki/
Set_(mathematics)
Shadrach, R. (2014). Introduction to Sets. Retrieved from
http://www.mathsisfun.com/sets/sets-introduction.html

LEARNING OUTCOMES
 INTRODUCTION
Did you notice that in real life, relationship exists between people and other
entities? For example „father of‰ is a relationship between two individuals.
Similarly, we may have a relationship „owner of‰ between people and car.
How about mathematics? In mathematics, this concept can be formalised by
using relation. Thus, we will discuss the concept of relation in this new topic.
You will be introduced to several geometric and algebraic methods for
representing relations between objects. Some discussion on the properties of
relations will be included too.
1. Describe the concept of relations between two sets;
2. Use the appropriate methods for representing relations;
3. Describe inverse relation;
4. Explain composition of relations;
5. Summarise four properties of relations on a set;
6. Draw a digraph to represent a relation; and
7. Distinguish partial order and equivalence relation.
T
To
op
pi
ic
c
2
2
 Relations

TOPIC 2 RELATIONS  29
CONCEPT OF RELATION
Firstly, let us learn the meaning of relation.
Now let us look at some examples of relations.
Example 2.1a
If X is a set of students
X = {Jimmy, Sheila, Shah, Zurai}
and Y is a set of courses
Y = {Computer Science, Math, Art, History}
a relation R between X and Y indicating „courses taken by the students‰ can be
written as
R = {(Jimmy, Computer Science), (Sheila, Math), (Jimmy, Art),
(Shah, History), (Shah, Computer Science), (Zurai, Math)}.
Since (Shah, History)R, we may write Shah R History.
We can represent a relation in Example 2.1a pictorially by using an arrow
diagram as follows (Figure 2.1).
2.1
Definition 2.1a: Let X and Y be two sets. A (binary) relation R from X to Y is
a subset of the Cartesian product X ×Y. If (x, y)R, we write x R y.
Definition 2.1b: The set
{x  X | (x, y) R for some y Y}
is called the domain of R. The set
{y Y | (x, y)R for some x X}
is called the range of R.

 TOPIC 2 RELATIONS
30
Figure 2.1: Relation between X and Y pictorially
Example 2.1b
Suppose A is a set of lecturers and B is a set of cars.
A = {Ahmad, Johan, Ravie, Zul}
B = {Iswara, Wira, Mercedes}
We can define a relation R between the two sets indicating „types of cars owned
by lecturers‰ as
R = {(Ahmad, Iswara), (Johan, Wira), (Johan, Mercedes), (Ravie, Wira)}
The pictorial representation of this relation is as follows:
Figure 2.2: Relation between A and B in pictorial representation
Example 2.1c
Suppose X = {2, 3, 4} and Y = {3, 4, 5, 6, 7}. If we define a relation R from X to Y by
(x, y)R, if x divides y (with zero remainder)
we obtain
R = {(2, 4), (2, 6), (3, 3), (3, 6), (4, 4)}
The domain of R is the set {2, 3, 4} and the range of R is the set {3, 4, 6}.

1. Suppose that
A = {Kota Bharu, Taiping, Ipoh, Seremban}
B = {Selangor, Kelantan, Perak}
Write a relation R between set A and set B which is defined by (x, y)
R if „x is a town in y .‰
2. A car manufacturer makes three different types of car frames and
two types of engines. List all possible models of cars.
Frame type: sedan (s), coupe (c), van (v)
Engine type: gas (g), diesel (d)
3. Suppose that
X = {a, b, c, d}
Y = {1, 2, 3, 4, 5}
For the relations R below, determine their domains and ranges.
(a) R = {(a, 1), (b, 2), (c, 3), (d, 4)}
(b) R = {(a, 2), (b, 4), (c, 1), (d, 5)}
(c) R = {(a, 4), (b, 2), (c, 4), (d, 2)}
4. Suppose that
X = {1, 2, 3, 4, 5}
Y = {3, 6, 9}
Let x  X, y  Y and write a relation R between set X and set Y
defined by:
(a) (x, y)  R if x < y
(b) (x, y)  R if y = 2x
ACTIVITY 2.1

32
INVERSE RELATION
Now let us move on to inverse relation. Firstly, let us learn its meaning.
Let us look at an example of inverse relation.
Example 2.2a
Let
X = {2, 3, 4} and Y = {3, 4, 5, 6}.
If we define a relation R from X to Y by
(x, y) R if x divides y
we obtain
R = {(2, 4), (2, 6), (3, 3), (3, 6), (4, 4)}
2.2
Definition 2.2a: Let R be a relation from X to Y. The inverse of R, denoted by R-1,
is the relation from Y to X defined by
R-1 = {(y, x) | (x, y)  R}
SELF-CHECK 2.1
I hope that you have understood the basic concepts of relation. Based on
your understanding, write a relation for the following items, G =
gender; and T = toys that kids love to play. You may present your idea
pictorially.

This relation can be presented pictorially as
Figure 2.3: Relation between X and Y pictorially
The inverse of the relation R is
R-1 = {(4, 2), (6, 2), (3, 3), (6, 3), (4, 4)}.
We can represent it pictorially as Figure 2.4.
Figure 2.4: Inverse relation, R-1 between X and Y pictorially
This relation can be described as „x is divisible by y.‰
Give the inverse for all the relations below:
(a) R = {(a, 6), (b, 2), (a, 1), (c, 1)}
(b) R = {(Suzi, Music), (Emmy, History), (Adri, Mathematics), (Emmy,
Chemistry)}
(c) R = {(2, 2), (5, 6), (1, 2), (7, 1), (9, 1)}
(d) R = {(8, 26), (21, 17), (10, a), (c, 45), (b, 3), (c, 3)}
(e) R = {(Blue, Car), (Red, Flower), (Black, Car), (White, Flower)}
ACTIVITY 2.2

34
COMPOSITION OF RELATIONS
Now let us learn composition of relations.
We can represent this relationship pictorially as shown in Figure 2.5.
Figure 2.5: R2 o R1
Let us look at an example of this relationship.
Example 2.3a
Suppose that we have two relations
R1 = {(1, 2), (1, 6), (2, 4), (3, 4), (3, 6), (3, 8)}
and
R2 ={(2, u), (4, s), (4, t), (6, t), (8, u)}
From Definition 2.3a, the relations can be presented pictorially as shown in
Figure 2.6.
2.3
Definition 2.3a: Let R1 be a relation from X to Y and R2 be a relation from Y to Z.
The composition of R1 and R2, denoted by R2 o R1, is the relation from X to Z
defined by
R2 o R1 = {(x, z) | (x, y)  R1 and (y, z)  R2 for some y  Y}

Figure 2.6: R2 o R1
The composition of these two relations is
R2 o R1= {(1, u), (1, t), (2, s), (2, t), (3, s), (3, t), (3, u)}
Figure 2.7: Composition of the two relations
Write the composition of R2 o R1 for the given relations
(a) R1 = {(4, 10), (8, 2), (6, 6)}
R2 = {(10, w), (2, z), (6, y)}
(b) R1 = {(Math, 3), (Art, 1), (Math, 1), (History, 4), (Chemistry, 2)}
R2 ={(1, Adri), (1, Amy), (2, Amin), (3, Shah), (4, Amy)}
(c) R1= {(Black, a), (Blue, a), (White, b), (Green, d), (Blue, c), (White, c)}
R2 = {(c, 2), (c, 6), (a, 4), (b, 4), (d, 6), (a, 8)}
(d) R1 = {(4, Black), (1, Red), (2, Blue), (3, White)}
R2 = {(Blue, Car), (Red, Flower), (Black, Car), (White, Flower)}
ACTIVITY 2.3

36
RELATIONS ON A SET
What can we say about relations on a set?
Let us look at an example of this.
Example 2.4a
Let R be the relation on
X = {1, 2, 3, 4, 5}
defined by (x, y)  R if x  y; x, y  X. Then
R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4),
(3, 5), (4, 4), (4, 5), (5, 5)}
The domain of R is the set {1, 2, 3, 4, 5} and the range of R is the set {1, 2, 3, 4, 5}.
Hence the domain and the range of R are both equal to X.
There are four properties of relations on a set. These properties are listed in
Figure 2.8.
Figure 2.8: Four properties of relations on set
These four properties are further discussed in the following subtopics.
2.4
Definition 2.4a: A (binary) relation R on a set X is a relation from X to X.

2.4.1 Reflexive
What does reflexive mean?
Example 2.4b
The relation R on X = {1, 2, 3, 4, 5} for Example 2.4a is reflexive because for all
elements x  X and (x, x)  R. Thus the reflexive elements are (1, 1), (2, 2), (3, 3),
(4, 4) and (5, 5).
Example 2.4c
Consider the following relations on {a, b, c}:
R1 = {(a, a), (a, b), (b, b), (b, c), (c, c)}
R2 = {(a, a), (a, b), (b, b), (a, c), (b, c)}
R3 = {(a, a), (b, b), (c, c)}
R4 = {(a, b), (a, c), (b, c)}
The relations R1 and R3 are reflexive because both relations contain all pairs of the
form (x, x).
The relations R2 and R4 are not reflexive because both relations do not contain all
pairs of the form (x, x).
2.4.2 Symmetric
When is a relation symmetric?
The following Example 2.4d shows you a symmetric relationship.
Definition 2.4b: A relation R on a set X is called reflexive if (x, x)  R for all x
X.
Definition 2.4c: A relation R on a set X is called symmetric if for all x, y  X,
and (x, y)  R, then (y, x)  R.

38
Example 2.4d
Consider the relation R on X = {a, b, c, d } given as follows:
R = {(a, d ), (b, c), (c, b), (d, a)}
The relation is symmetric because for all x and y, when (x, y )  R, (y, x )  R.
In this case (a, d ) and (d, a), (b, c) and (c, b) are all in R.
2.4.3 Antisymmetric
What happens when a relation is antisymmetric?
Let us examine Example 2.4e.
Example 2.4e
Consider the relation R on X = {a, b, c, d } which is given as follows:
R = {(a, b), (b, c), (c, d )}
The relation is antisymmetric because (a, b) R and (b, a) R, then a b. It is
also the same for (b, c) and (c, d ).
2.4.4 Transitive
What does a transitive relationship mean?
Let us look at Example 2.4f.
Definition 2.4e: A relation R on a set X is called transitive if for all x, y, z
 X, when (x, y)  R, (y, z)  R and (x, z)  R.
Definition 2.4d: A relation R on a set X is called antisymmetric if for all x, y
 X; when (x, y)  R and (y, x)  R, then x ≠ y.

Example 2.4f
Consider the relation R on X = {a, b, c, d } which is given as follows:
R = {(a, b), (a, d ), (a, c), (b, c), (b, d ),(c, d )}
The relation R is transitive because for all x, y, z where (x, y) and (y, z) R, (x,
z) R.
For example, (a, b) R, (b, c) R, and (a, c) R;
(a, b) R, (b, d ) R, and (a, d ) R; and
(a, c) R, (c, d ) R and (a, d ) R.
DIGRAPH
Did you realise that one way to picture a relation on a set is to draw its digraph?
A digraph consists of vertices to represent the elements of X and edges to
represent the relation between the elements. Let us check out some examples of
digraphs.
2.5
1. Write each of the relations below as relations on sets
(a) The relation R on {1, 2, 3, 4} defined by (x, y ) R if x2 y
(b) The relation R on {1, 2, 3, 4, 5} defined by (x, y ) R if y = 2x
(c) Relation R on the set {1, 2, 3, 4, 5} defined by the rule (x, y ) R
if 3 divides x – y.
(d) Relation R on the set {1, 2, 3, 4, 5} defined by the rule (x, y ) R
if x + y6.
(e) Relation R on the set {1, 2, 3, 4, 5} defined by the rule (x, y ) R
if x = y – 1.
2. Is the relation in exercise 1(a) to (e) reflexive, symmetric,
antisymmetric or transitive?
ACTIVITY 2.4

40
Example 2.5a
The digraph for the relation
R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4),
(3, 5), (4, 4), (4, 5), (5, 5)}
can be drawn as follows:
Figure 2.9: Digraph for R
Example 2.5b
The relation R on X = {a, b, c, d }
R = {(a, d ), (b, c), (c, b), (d, a)}
is given by the digraph as follows:
Figure 2.10: Relation of symmetric

Did you know that we can determine the characteristics of a relation by using
digraphs? These characteristics are summarised in Table 2.1.
Table 2.1: Characteristics of a Relation and Digraph
Set Relation Digraph
Reflexive Has a loop at every vertex.
Symmetric Has the property that whenever there is a directed edge from v to
w, there is also a directed edge from w to v.
Antisymmetric Has the property that between any two vertices there is at most one
directed edge.
Transitive Has the property that whenever there are directed edges from x to y
and from y to z, there is also a directed edge from x to z.
PARTIAL ORDER
What does partial order mean?
Let us look at some examples of partial orders.
2.6
Definition 2.6a: A relation R on a set X is called a partial order if R is reflexive,
antisymmetric and transitive.
Draw the digraph of the following relations and identify the
properties of relations on set:
(a) R = {(1, 2), (2, 1), (3, 3), (1, 1), (2, 2)} on X = {1, 2, 3}
(b) R = {(1, 2), (2, 3), (3, 4), (4, 1)} on {1, 2, 3, 4}
(c) R on {1, 2, 3,4} defined by (x, y)  R if x2  y
(d) R = {(a, 3), (b, 1), (c, 4), (d, 1)} from {a, b, c, d} to {1, 3, 4}
ACTIVITY 2.5

42
Example 2.6a
Consider the relation R on the set X = {1, 2, 3, 4, 5} defined by
(x, y) R if x divides y
So R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 4) (3, 3), (4, 4), (5, 5)}
R is reflexive since (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) are in R.
R is antisymmetric since (2, 1), (3, 1), (4, 1), (5, 1) and (4, 2) are not in R.
R is transitive since (1, 2), (2, 4) and (1, 4) are in R.
Thus R is a partial order.
Example 2.6b
In general, a relation R on a set of positive integers defined by
(x, y) R if x divides y
is a partial order.
If R is a partial order on a set X, we can denote x y to indicate that (x, y)R.
Then we have comparable and incomparable. What do they mean?
Example 2.6c
The less than or equals relation on the positive integers is a total order since, if x
and y are integers, either x y or y x.
Definition 2.6b: Suppose that R is a partial order on a set X. If x, y  X and
either x  y or y  x, we say that x and y are comparable. Otherwise, we say
that x and y are incomparable.
Definition 2.6c: If every pair of the elements in X is comparable, we call R a
total order.

EQUIVALENCE RELATION
What does equivalence relation mean?
Let us look at two examples which illustrate equivalence relation.
Example 2.7a
Consider the relation R on {1, 2, 3, 4, 5} defined as
R = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (5, 1),
(5, 3), (5, 5)}
R is reflexive because (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) R.
R is symmetric because whenever (x, y) is in R, (y, x) is also in R.
R is transitive because whenever (x, y) and (y, z) are in, (x, z) is also in R.
Thus, R is an equivalence relation on {1, 2, 3, 4, 5}.
Example 2.7b
The relation R as shown in the Example 2.6a is not an equivalence relation
because R is not symmetric.
2.7
Determine whether each of the relations defined below on the set of
positive integers is a partial order or not.
(a) (x, y)  R if x = y2
(b) (x, y)  R if x > y
(c) (x, y)  R if x  y
(d) (x, y)  R if x = y
(e) (x, y)  R if 3 divides x – y
ACTIVITY 2.6
Definition 2.7a: A relation that is reflexive, symmetric and transitive on a
set X is called an equivalence relation on X.

44
• The relation R between set X and Y is said „x is related to y‰, and denoted by
x R y when (x, y)  R.
• The relation can be presented pictorially by using an arrow diagram.
• Let R be a relation from X to Y. The inverse of R, denoted by R-1, is the
relation from Y to X defined by R-1 = {(y, x) | (x, y) R}
1. Determine whether the given relation is an equivalence
relation on {1, 2, 3, 4, 5}.
(a) {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 3), (3, 1)}
(b) {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 3), (3, 1), (3, 4), (4, 3)}
(c) {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 5), (5, 1), (3, 5), (5, 3), (1, 3),
(3, 1)}}
For the following questions, x, y  {1, 2, 3, 4, 5}.
(d) {(x, y ) | 1  x  5, 1 y  5}.
(e) {(x, y ) | 4 divides x – y}.
(f) {(x, y ) | 3 divides x + y}.
(g) {(x, y ) | x divides 2 – y}.
2. Determine whether the given relation is an equivalence relation
on the set of all people.
(a) {(x, y ) | x and y are the same height}.
(b) {(x, y ) | x and y have, at some time, lived in the same
country}.
(c) {(x, y ) | x and y have the same first name}.
(d) {(x, y ) | x is taller than y }.
(e) {(x, y ) | x and y have the same parents}.
(f) {(x, y ) | x and y have the same colour hair}.
ACTIVITY 2.7

• Let R1 be a relation from X to Y and R2 be a relation from Y to Z. The
composition of R1 and R2, denoted by R2 o R1, is the relation from X to Z,
defined by R2 o R1 = {(x, z) | (x, y)R1 and (y, z)R2 for some y Y }
• A (binary) relation R on a set X is a relation from X to X.
• There are four properties of relations on a set namely reflexive, symmetric,
antisymmetric and transitive.
• The relation R between two sets is reflexive when (x, x )  R for all x  X.
• The relation R between two sets is symmetric when (x, y )  R then (y, x)  R
for all x, y  X.
• The relation R between two sets is antisymmetric when (x, y )  R but (y, x) 
R for all x, y  X.
• The relation R between two sets is transitive when (x, y )  R and (y, z)  R
then (x, z)  R for all x, y, z  X.
• The partial order is a relation R with properties of reflexive, antisymmetric
and transitive.
• The equivalence relation is a relation R with properties of reflexive,
symmetric and transitive.
Antisymmetric relations
Composition of relations
Digraph
Domain
Equivalence relations
Inverse relations
Partial order
Range
Reflexive relations
Relations
Symmetric relations
Transitive relations

46
Application to Graph theory. Retrieved from http://aix1.uottawa.ca/
~jkhoury/graph.htm
Graph matrices. (2010). Retrieved from http://compalg.inf.elte.hu/~tony/
Oktatas/TDK/FINAL/ Chap%2010.PDF
Pearson.
McGraw Hill.

LEARNING OUTCOMES
 INTRODUCTION
Are you aware that the concept of functions is very important in discrete
mathematics? In fact, functions play important roles in information technology,
such as, to represent the computational complexity of algorithms. In addition,
there are many programs designed to calculate the values of functions.
What does a function mean? A function is basically a special kind of relation that
has exactly one output for every possible input in the domain. So, most of the
terminologies that have been introduced in Topic 2 will be reused in this topic.
1. Describe the concept of function;
2. Represent functions by using graphical representations;
3. Identify three types of functions;
4. Discuss inverse of a function;
5. Apply the operation on function composition; and
6. Distinguish between binary and unary operators.
T
To
op
pi
ic
c
3
3
 Function

 TOPIC 3 FUNCTION
48
CONCEPT OF FUNCTION
Before we go further, let us learn the concept of function.
Let us look at some examples.
Example 3.1a
The relation
f = {(1, a), (2, b), (3, a)}
from X = {1, 2, 3} to Y = {a, b, c} is a function from X to Y.
By using an arrow diagram, this relation can be presented as:
Figure 3.1: The relation f
The domain of f is X and the range of f is {a, b }.
The range of a relation R is the set {y Y |(x, y ) R for some x  X }
3.1
Definition 3.1a: A function f from X to Y is a relation from X to Y having the
following properties:
1. For each x X
 , there is exactly one y Y
 with  
,
x y f
 .
2. The domain of f is X and the range of f is  
 
,
y x y f
 .
A function from X to Y is sometimes denoted by f : X  Y

TOPIC 3 FUNCTION  49
Example 3.1b
The relation is represented as
f(x) = x2 ; x ∈ R
This relation is a function and the domain is the set of all real numbers.
The range of f is the set of all non-negative real numbers.
If f is a function f: X2, given a value in set X, we can obtain the value in set Y. This
process is called function application.
Example 3.1c
We can apply the function given in Example 3.1b as follows:
f (2) = 4,
f (-2) = 4,
f (8) = 64.
1. How can you relate the concept of function and apply it to your
daily life? Give an example.
2. Determine whether the relation R below is a function from
X = {1, 2, 3, 4} to Y = {a, b, c, d }.
If it is a function, find its domain and range, and then draw
the arrow diagram.
(a) R = {(1, a), (2, a), (3, c), (4, b)}
(b) R = {(1, c), (2, a), (3, b), (4, c), (4, d )}
(c) R = {(1, c), (2, d ), (3, a), (4, b)}
(d) R = {(1, d ), (2, d ), (4, b)}
(e) R = {(1, b), (2, b), (3, b), (4, b)}
ACTIVITY 3.1

50
GRAPH OF A FUNCTION
Are you aware that another way to visualise a function is to draw its graph? Let
us look at some examples on how to do it.
Example 3.2a
The graph of the function
f(x) = x2,
has the value as in Table 3.1 and the graph is shown in the Figure 3.2.
Table 3.1 The value of the function f(x) = x2
x -2 -1 0 1 2
f(x) = x2 4 1 0 1 4
3.2
3. Determine whether the relation R from A to B is a function.
(a) A = The set of all learners at Open University Malaysia
B = The set of courses offered by Open University Malaysia
(b) A = A set of people in Putrajaya
B = The set of IC numbers
4. Let g = {(1, a), (2, c), (3, c)} be a function from X = {1, 2, 3} to
Y = {a, b, c, d }. Apply the function to find the value of
(a) g(1)
(b) g(2)
5. Let   3, .
f x x x Z
   Apply the function to find the value of
(a) f (5)
(b) f (-1)

Figure 3.2: The graph of the function f(x) = x2
Example 3.2b
The graph of the function R = {(1, 1), (1, 3), (2, 2), (3, 0)} is as shown in Figure 3.3.
Figure 3.3: The graph of the function R

52
TYPES OF FUNCTIONS
Did you know that there are three types of functions? They are shown in Figure 3.4.
Figure 3.4: Three types of functions
These three types of functions are further discussed as follows.
3.3.1 Injective
What does injective mean?
3.3
1. Draw a graph for the function f (x) = x2 + 1 where x  0. Do you get
the parabola shape on your graph?
2. Represent the functions below by using graphical representation.
 
1 2 2
( ) ; ,
f x x x
   
 
3
3 3
( ) ; ,
f x x x
  
3. Represent the functions R below by using graphical representation.
(a) R = {(1, 2), (2, 2), (3, 6), (4, 4)}
(b) R = {(1, 3), (2, 4), (3, 5), (4, 1)}
ACTIVITY 3.2
Definition 3.3a: A function f from X to Y is said to be one to one (or
injective) if for each y Y, there is at most one x  X with f (x) = y.

The condition given in the definition above for a function to be one to one is
equivalent to: if x, xÊ  X and f(x) = f(xÊ), then x = xÊ.
If a function from X to Y is one to one, each element in Y in its arrow diagram
will have at most one arrow pointing to it (see Figure 3.5). If a function is not
one-to-one, some element in Y in its arrow diagram will have two or more
arrows pointing to it. Let us look at an example.
Example 3.3a
The function f = {(1, b), (3, a), (2, c)} from X = {1, 2, 3} to Y = {a, b, c, d }
is one-to-one.
Figure 3.5: One-to-one relationship
Example 3.3b
The function f(x) = x + 1; x  [-2, 2] is one-to-one.
x -2 -1 0 1 2
f (x) = x + 1 -1 0 1 2 3
Example 3.3c
The function R = {(1, a), (2, b), (3, a)} is not a one-to-one function because a has
two values.

54
3.3.2 Surjective
What is surjective?
If the function from X to Y is onto, each element in Y in its arrow diagram will
have at least one arrow pointing to it (see Figure 3.6). Let us look at some
examples.
Example 3.3d
The function f = {(1, D), (2, B), (3, C), (4, C)} is onto on {D, B, C}.
Figure 3.6: Onto relationship
Example 3.3e
Define :
f R R
 by the rule   5 2
f x x
  for all x R
 . Prove that f is onto.
Proof: Let y R
 . (We need to show that x
 in R such that  
f x y
 )
If such a real number exists, then 5 2
x y
  and
2
5
y
x

 . x is a real number
since sums and quotients (except for division by 0) of real numbers are real
numbers. It follows that
 
2
5 2
5
2 2
y
f x
y
y

 
 
 
 
  

Hence, f is onto.
Definition 3.3b: If f is a function from X to Y and the range of f is Y, f is
said to be onto Y (or an onto function or a surjective function).

3.3.3 Bijective
How about bijective? What does it mean?
Let us look at two examples of this.
Example 3.3f
The function f = {(1, a), (2, b), (3, c)} from X = {1, 2, 3} to Y {a, b, c } is one-to-one
and onto. So it is bijective.
Example 3.3g
The function f = {(1, a), (2, c), (3, b)} from X = {1, 2, 3} to Y = {a, b, c } is
one-to-one and onto Y. Thus, it is bijective.
The arrow diagram for this function is shown in Figure 3.7.
Figure 3.7: One-to-one and onto Y
Definition 3.3c: A function that is both one-to-one and onto is called a
bijective function.

56
INVERSE OF A FUNCTION
Suppose that f is one to one, onto function from X to Y. It can be shown that the
inverse relation
{(y, x)|(x, y)  f}
is a function from Y to X. This new function, denoted f–1 is called f inverse.
Given the arrow diagrams of a one-to-one, onto function f from X to Y, we can
obtain the arrow diagram for f-1 simply by reversing the direction of each arrow.
3.4
1. Show that if f is a one-to-one, onto function from X to Y, then
{(y, x) | (x, y)  f}
is a one-to-one, onto function from Y to X.
2. In each part, sets A and B and a function from A to B are given.
Determine whether the function is one-to-one or onto (or both or
neither).
(a) A = {1, 2, 3, 4} = B;
f = {(1, 1), (2, 3), (3, 4), (4, 2)}
(b) A = {1, 2, 3, 4}; B = {a, b, c, d };
f = {(1, a), (2, b), (3, c)}
(c) A = B = Z ; f (a) = a –1
(d) A = {1.1, 7, 0.06}; B = {p, q };
f = {(1.1, p), (7, q), (0.06, p)}
3. Let f be a function from A to B. Determine whether each function f
is one-to-one and whether it is onto or not.
(a) A = R, B = {x|x is a real and x  0}; f(a) =|a|
(b) A = R ╳R, B = R; f ((a, b)) = a
(c) A = B = R ╳R; f ((a, b)) = (a + b, a – b)
(d) A = R , B = {x|x is a real and x  0}; f(a) = a2.
ACTIVITY 3.3

Example 3.4a
For the function f in f = {(1, a), (2, c), (3, b)} from X = {1, 2, 3} to Y = {a, b, c } is one-
to-one and onto Y.
f–1 = {(a, 1), (c, 2), (b, 3)}
The arrow diagram for f–1, where f is the function is shown in the following
Figure 3.8.
Figure 3.8: The inverse function
1
f 
Example 3.4b
The function
f(x) = 2x
is one to one function from the set R of all real numbers onto the set R+ of all
positive real numbers. We will derive a formula for f-1(y).
Suppose that (y, x) is in f-1, that is
f-1 (y) = x
then (x, y)  f. Thus
y = 2x.
By definition of logarithm,
log2 y = x
Combining (f-1 (y) = x) and (log2 y = x), we have
f-1 (y) = x = log2 y

58
COMPOSITION OF FUNCTIONS
Since functions are special kinds of relations, we can form the composition
of two functions. Specifically, suppose that g is a function from X to Y and f is a
function from Y to Z. The resulting function from X to Z is called the composition
of f with g and is denoted by f o g.
Example 3.5a
Given g = {(1, a), (2, a), (3, c)} a function from X = {1, 2, 3} to Y = {a, b, c}, and f =
{(a, y), (b, x), (c, z)}, a function from Y to Z = {x, y, z}.
The composition function from X to Z can be represented by an arrow diagram as
shown in Figure 3.9.
Figure 3.9: Composition function from X to Z
So,
f o g = {(1, y), (2, y), (3, z)}
The application of f o g on x can be written as (f o g)(x) or alternatively as f(g(x)).
3.5
Each of the functions below is one-to-one. Find the inverse function for
each of them.
(a) f(x) = 4x + 2 (f) f(x) = 6+27x1
(b) f(x) = 3x (g) f(x) = 6x9
(c) f(x) = 3 log2 x (h) f(x) = 3x2
(d) f(x) = 3 + 1/x (i) f(x) = 2x34
(e) f(x) = 4x35
ACTIVITY 3.4

Example 3.5b
If f(x) = log3 x and g(x) = x4,
f(g(x)) = log3 (x4 ) , g(f(x)) = (log3 x) 4
Composition sometimes allows us to decompose complicated functions into
simpler functions.
Example 3.5c
The function
 
f x = sin2x
can be decomposed into the functions
  
g x x , h(x) = sin x, w(x) = 2x
and we can write
   
 
 

f x g h w x
ACTIVITY 3.5
1. Given f (n) = 3n + 2 and g(n) = 2n – 1. Find a composition of g o f,
and how do you pronounce the answer for g o f?
2. Let f and g be functions from the positive integers to the positive
integers defined by the equations
f(n) = 2n + 1
g(n) = 3n – 1
Find the following function compositions:
(a) f o f
(b) g o g
(c) f o g
(d) g o f

60
BINARY AND UNARY OPERATORS
Lastly, let us learn binary and unary operators; firstly, its definition.
Let us look at the following example.
Example 3.6a
Let X = {1, 2, ⁄.}. If we define f(x, y) = x + y, then f is a binary operator on X.
A unary operator of a set X associates each element of X with one element in X.
Example 3.6b
Let E be a universal set and X is a set. If we define
f(X) = X , X  P(E),
Then f is a unary operator on P(E).
3.6
3. Let f and g be functions from the positive integers to the positive
integers defined by the equations
f(x) = 2x
g(x) = x2
Find the following function compositions:
(a) f o f
(b) g o g
(c) f o g
(d) g o f
Definition 3.6a: A function from X x X into X is called a binary operator on X.
Definition 3.6b: A function X into X is called a unary operator on X.

• The function f from X to Y is a relation from X to Y, if for each element in X,
there is exactly one element in Y.
• A function can be drawn into a graphical representation.
• Three types of functions are injective, surjective and bijective.
• The function f from X to Y is said to be one to one (or injective), if each element
in Y in its arrow diagram will have at most one arrow pointing to it.
• The function f from X to Y is said to be onto (or surjective), if each element in Y
in its arrow diagram will have at least one arrow pointing to it.
• The function f from X to Y is said to be bijective function, if it has both one-
to-one and onto function.
• Suppose that f is one to one, onto function from X to Y. It can be shown that
the inverse relation
{(y, x)|(x, y) ∈ f }
is a function from Y to X. This new function, denoted f–1 is called f inverse.
1. A binary operator f in the set X is commutative if f(x, y) = f(y, x)
for all x, y  X. State whether the given function f is a binary
operator on the set X. If f is not a binary operator, state why. State
whether or not each binary operator is commutative.
(a) f(x, y) = x + y, X = {1, 2, ⁄.}
(b) f(x, y) = x – y, X = {1, 2, ⁄.}
(c) f(x, y) =x/y, X = {0, 1, 2, ⁄.}
(d) f(x, y) = x2 + y2 – xy, X = {1, 2, ⁄.}
2. Give an example of a unary operator (different from f(x) = x, for
all x) on the given set.
{⁄, -2, -1, 0, 1, 2, ⁄}
ACTIVITY 3.6

62
• The operation on function composition happens when g is a function from X
to Y and f is a function from Y to Z. Then the resulting function from X to Z is
the composition of f with g and is denoted by f o g.
• A function from X  X into X is called a binary operator on X.
• A function X into X is called a unary operator on X.
Bijective
Binary operators
Composition of functions
Functions
Injective
Inverse of functions
One-to-one
Onto
Surjective
Unary operators
Discrete Mathematics/Functions and relations. (2015). Retrieved from http://
en.wikibooks.org/wiki/Discrete_Mathematics/Functions_and_relations
Farlex Inc. (2015). The Free Dictionary: Function. Retrieved from
http://www.thefreedictionary.com/function
Johnsonbaugh, R. (2014). Discrete mathematics (7th ed.). Edinburgh Gate: Pearson.
NCS Pearson. (2014). Functions in Discrete Mathematics. Retrieved from
http://math.tutorcircle.com/discrete-math/functions-in-discrete-
mathematics.html
McGraw Hill.

LEARNING OUTCOMES
 INTRODUCTION
What is the function of sequences? Sequences are used to represent an ordered
list of elements. A list of the letters as they appear in a word (or normally called a
string) is an example of a sequence. Another example is the word „form‰ and
„from‰; they are two different words although both of them consist of the same
letters. Thus, the concept of sequence and strings will be the discussion of this
topic.
1. Describe the concept of sequence;
2. Apply the operations on sequences;
3. Describe the concept of strings; and
4. Apply the operations on strings.
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 Sequence and
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 TOPIC 4 SEQUENCE AND STRINGS
64
SEQUENCE
What can you say to define sequence? A sequence is a list in which order is taken
into account. It is a special type of function. The formal definition of a sequence is
as follows:
Let us look at some examples of sequence.
Example 4.1a
The ordered list 2, 4, 6, ⁄. 2n⁄.. is a sequence. The first element is 2, the second
element is 4 and so on. The nth element is 2n. If we let s denote this sequence, we
have
Figure 4.1: Ordered list sequence
Example 4.1b
KLIA Shuttle Inc. charges RM1 for the first km and 50 cents for each additional
km. In general, the cost cn of travelling n km is 1.00 (the cost of travelling the first
km) plus 0.50 times the number (n – 1) of additional km. That is,
cn = 1 + 0.5 (n – 1)
For example:
c1 = 1 + 0.5 (1 – 1)
= 1 + 0.5 (0)
= 1
c5 = 1 + 0.5 (5 – 1)
= 1 + 0.5 (4)
= 3
4.1
Definition 4.1a: A sequence whose smallest index is 1 is a function whose
domain is either the set of all positive integers or a set of the form {1 ⁄. n}.
If s is a sequence, we denote the first element as s1, the second element as s2
and so on. In general, sn denotes the nth element.

TOPIC 4 SEQUENCE AND STRINGS  65
The list fare in this example is in sequence.
A sequence can have repetitions.
Example 4.1c
The ordered list a, a, b, a, b⁄. is a sequence. The first element of the sequence is
a, the second element of the sequence is a and so on. If we denote this sequence,
we have
Figure 4.2: Ordered list
An alternative notation for the sequence s is sn. Here, s or sn denotes the
entire sequence s1, s2, s3 , ⁄ We use the notation sn to denote the single, nth
element of the sequence s.
A sequence sn is defined by the rule sn = n2  nthe first five terms of this
sequence are
Figure 4.3: Sequence of sn by the rule sn = n2 – 1
The 55th term is
s55 = 552 – 1 = 3024
Example 4.1d
A sequence u is defined by the rule un, which is the nth letter in the word
„digital‰. Then u1 = d, u2 = u4 = i and u7 = l. This sequence is a finite sequence.
Word d i g i t a l
Sequence u1 u2 u3 u4 u5 u6 u7
Figure 4.4: Sequence of un of the word „digital‰

66
Example 4.1e
If x is the sequence defined by
 
1
3
n
x
n
1 n  4
The elements of x are
1 1 1
3,1, , ,
3 9 27
4.1.1 Types of Sequence
There are two important types of sequence, namely, increasing sequence and
decreasing sequence.
Example 4.1f
The sequence s
2, 4, 6, ⁄
is increasing since sn = 2n  2(n+1) = sn+1 for all n.
Example 4.1g
The sequence s
3, 5, 5, 7, 8, 8, 13.
is increasing since sn  sn+1 for all n.
Definition 4.1b: A sequence s is increasing if sn  sn+1 for all n.
Definition 4.1c: A sequence s is decreasing if sn  sn+1 for all n.

Example 4.1h
The sequence X
1 1 1
3,1, , ,
3 9 27
,⁄
is decreasing since 

   1
1
1 1
3 3
n n
n n
X X for all n.
4.1.2 Subsequence
One way to form a new sequence from a given sequence is to retain only certain
terms of the original sequence and maintaining the order terms in the given
sequence. The resulting sequence is called a subsequence of the original sequence.
Examples for subsequence are as follows:
Example 4.1i
The sequence
b, c
is a subsequence of the sequence
a, a, b, c, q
Notice that the sequence
c, b
is not a subsequence of the sequence.
Definition 4.1d: Let  sn  be a sequence defined for n = m, m + 1, m + 2, ⁄
and let n1, n2, n3, ⁄ be an increasing sequence satisfying nk , nk+1, nk+2 for
all k, whose values are in the set { m, m + 1, m + 2, ⁄ }. We call the sequence
k
n
s a subsequence of n
s .

68
Table 4.1 illustrates this example further.
Table 4.1: Example 4.1j
Example 4.1j
The sequence
2, 4, 8, 16
is a subsequence of the sequence
2, 4, 6, 8, 10, 12, 14, 16.
Sequence a, a b, c q
Subsequence of sequence b, c
Therefore, c, b is not a subsequence of the sequence
ACTIVITY 4.1
What are strings? Are strings often restricted to sequence?
SELF-CHECK 4.1
1. The sequence s is defined by c, d, d, c, d, c
(a) Find s1
(b) Find s4
2. The sequence k is defined by
kn = 2n – 1, n > 1
(a) Find k3
(b) Find k100
(c) Find k7
(d) Find k2077
(e) Is k increasing or decreasing?

SEQUENCE OPERATION
Let us learn on sequence operation.
Operations in sequence are shown in the following examples.
4.2
3. The sequence r defined by
  
3.2 4.5 , 0
n n
n
r n
(a) Find r0
(b) Find r1
(c) Find r2
(d) Find r3
(e) Find a formula for rp
(f) Find a formula for rn-2
(g) Show that n
r satisfies
 
  
1 2
7 10 , 2
n n n
r r r n
(h) Is r increasing or decreasing?
4. Define  sn by the rule
sn = 2n – 1, n > 1
The subsequence of s obtained by taking first, third, fifth terms
(a) List the first seven terms of s.
(b) List the first seven terms of the subsequence.
(c) Find a formula for the expression nk of Definition 4.1d.
(d) Find a formula for the k th term of the subsequence.
Definition 4.2a: If
n
i i=m
a is a subsequence, we define the sum and product
of terms in the sequence as
Sum of terms:  
    
1 ...
n
i m i m m n
a a a a
Product of terms:  
    
1 ...
n
i m i m m n
a a a a

70
Example 4.2a
Let a be a sequence defined by an = 2n, n1. Then

   
1 1 2 3
n
i i
a a a a
= 2(1) + 2(2) + 2(3)
= 2 + 4 + 6
= 12

   
3
1 1 2 3
i i
a a a a
= 2(1)  2(2)  2(3)
= 2  4  6
= 48
Example 4.2b
The geometric sum
a + ar + ar 2 + ⁄ + ar n
can be rewritten compactly using the sum notation as


n
i
i 0
ar
Example 4.2c
Let a be the sequence defined by the rule an = 2( 1)n, where n >1. Find a formula
for the sequence s defined by

  0
n
n i i
s a
We find that
n
s = 2(1)1 + 2(-1)2 + 2(-1)3 + ⁄ + 2(-1)n
= 2 + 2 ⁄ + (-1)n 2 =
0 if is even
2 if is odd
n
n





1. The sequence g is defined by
gn = n2 – 3n + 3, n > 1
(a) Find
4
i
=1
i
g

(b) Find
3
5
i
i
g


(c) Find
=1
6
k
k
g

(d) Find
2
1
i
i
g


(e) Find
3
1
i
i
g


2. The sequence v is defined by vn = n(1)n
(a) Find
1
4
i
i
v


(b) Find
1
10
i
i
v


(c) Find a formula for the sequence c defined by
1
n
n i
i
c v

 
(d) Find a formula for the sequence d defined by
1
n
n i
i
d v

 
3. Rewrite the sum
2
1
-
n i
n
i
v v


Replacing the index i by k, where i = k+1
ACTIVITY 4.2

72
STRING
The following is the definition for string.
Example 4.3a
Let X = {a, b, c}. If we let
q1 = b, q2 = a, q3 = a, q4 = c
We obtain a string over X. This string is written baac.
Figure 4.5: String with order
Since a string is a sequence, order is taken into account. For example, the string
baac is different from the string acab.
Repetition in a string can be specified by superscripts. For example, the string
bbaaac may be written b2a3c.
The string with no elements is called the null string and is denoted as 
4.3
Definition 4.3a: A string over X, where X is a finite set, is a finite sequence of
elements from X.
Definition 4.3b: We let X* denotes the string of all strings over X,
including the null string, and we let X+ denote the set of all nonnull strings
over X.

Example 4.3b
Let X = {a, b}. Some elements in X* are
, a, b, abab, b20a5 ba
Example 4.3c
If  = aabab and  = a3b4a32, then
|| = 5 and || = 39
If  and  are two strings, the string consisting of  followed by , written ,
is called the concatenation of  and .
Example 4.3d
If t = aab and e = cabd, then
te = aabcabd, et = cabdaab, t = t = aab,
t = t = aab.
Definition 4.3c: The length of a string is the number of elements inhe
length ofis denoted by||
1. Let X={a, c, e, i, m, s, t }. Find the sequence if the string contains the
word „mathematics‰.
2. Suppose we have three strings as follows
 = baab,  = caaba,  = bbab
(a) Write the string , , , 3
2
, 2
, , 2

(b) Compute the value of ||, ||, ||, ||
ACTIVITY 4.3

74
• A sequence is a list in which order is taken into account, such as, if s is a
sequence, we denote the first element as s1, the second element as s2 and nth
element denotes as sn. The sequence is increasing when sn  sn+1 for all n. The
sequence is decreasing when sn+1  sn for all n. The certain terms of the
original sequence is called a subsequence. For example, sequence A contains
s1, s2, s3, s4 and s5 while s2, s3 and s4 is a subsequence of A.
• The operation on sequence involves the sum and product of terms in the
sequence.
• A string is a finite sequence of elements which are not necessarily distinct
elements. For example, abaa is the string with four elements and baaa is also
the string with four elements but they are two different strings.
• The operations on strings includes length of a string and concatenation.
Concatenation
Decreasing
Increasing
Length of a string
Nonnull string
Null string
Sequence
String
Subsequence
Pearson.
McGraw Hill.

LEARNING OUTCOMES
 INTRODUCTION
Are you aware that logic is the foundation of science and mathematics? We use
logical methods in mathematics to prove the validity of mathematical
statements. As for natural and physical sciences, logic is used to draw
conclusions from experiments or observations. How about in information
technology? In information technology, logic can be used to verify the
correctness of computer programs.
Did you know that there are a few types of logic? In this topic, we will deal with
the simplest form of logic, namely, the propositional logic. This type of logic deals
with propositions and how to reason on propositions.
1. Explain the concept of proposition;
2. Formulate a proposition in words into a symbolic expression for
conditional and biconditional proposition;
3. Solve the truth value of compound propositions using truth tables or
laws of logic; and
4. Apply operations on compound propositions.
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 Propositional
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 TOPIC 5 PROPOSITIONAL LOGIC
76
PROPOSITION
Let us learn the basis of propositions. According to Poly (1945), in order to
translate a sentence from English into French, two things are necessary. First, we
must understand thoroughly the English sentence. Second, we must be familiar
with the forms of expression peculiar to the French language. The situation is
very similar when we attempt to express in mathematical symbols a condition
proposed in words. First, we must understand thoroughly the condition. Second,
we must be familiar with the forms of mathematical expressions.
Now, let us learn the meaning of proposition. Do you have any idea?
Example 5.1a
Which of the following are propositions?
(a) The Earth is round.
(b) Malaysia is a country in South East Asia.
(c) Do you speak English?
(d) 6 + 2x = 5.
(e) Take two panadol tablets.
(f) The temperature in Malaysia is between 28F to 38F.
Solution:
Question (a) and (b) are statements that happen to be true.
Question (c) is a question, so it is not a statement that is either true or false.
Question (d) is not a statement, since it is true or false depending on the value of x.
Question (e) is not a statement, it is a command.
Question (f) is a declarative sentence whose truth or falsity we do not know at this
time; however, we can (in principle) determine if it is true or false; so it is a
statement.
5.1
Definition 5.1a: A statement that is either true or false, but not both, is called
a proposition. It is expressed as a declarative sentence.

TOPIC 5 PROPOSITIONAL LOGIC  77
In propositional logic, a proposition is represented by using a lower case letter,
such as p, q or r. For example, we can use the notation
p : The Earth is round
to define p as a proposition „The Earth is round.‰
5.1.1 Conjunction and Disjunction
Let us look at conjunctions first.
We can describe the values of the conjunction of p and q by using a truth
table as shown in Table 5.1.
Table 5.1: Conjunction of p and q
p q p ∧ q
T
T
F
F
T
F
T
F
T
F
F
F
Now let us move on to disjunction. What does it mean?
Definition 5.1b: Let p and q be propositions. The conjunction of p and q,
denoted p  q, is the proposition p and q. The value of this proposition is
true (T ) if both p and q are true, otherwise the proposition is false (F ).
Definition 5.1c: Let p and q be propositions. The disjunction of p and q,
denoted p  q, is the proposition p or q. The value of this proposition is
true (T ) if p or q is true (T ). It is false if both p and q are false (F ).

78
The following Table 5.2 is the truth table for the disjunction of p and q.
Table 5.2: Disjunction of p and q
p q p  q
T
T
F
F
T
F
T
F
T
T
T
F
Propositions, such as, p  q and p  q that result from combining propositions
are called compound propositions. Let us look at Example 5.1b.
Example 5.1b
If
p: The Earth is round,
q: A decade is 100 years,
Then the conjunction of p and q is
p  q: The Earth is round and a decade is 100 years
Since p is true and q is false, p  q is false.
The disjunction of p and q is
p  q: The Earth is round or a decade is 100 years.
Since p is true and q is false, p  q is true.
5.1.2 Negation
What is negation?
Definition 5.1d: The negation of p, denoted p is the proposition not p.

Table 5.3 shows you the truth table for the negation p.
Table 5.3: Negation p
p p
T F
F T
Let us check out an example for negation.
Example 5.1c
If p : The earth is round.
The negation of p is the proposition
p : The earth is not round.
Since p is true, p
 is false.
ACTIVITY 5.1
1. Determine whether each of the statements given is a proposition.
If the statement is a proposition, write its negation.
(a) Give me a cup of tea.
(b) 2 + 5 = 19.
(c) For some positive integer n, 19340 = n  17.
(d) The difference of two primes is always greater than 1.
2. Write the truth value for each of the propositions below:
p = F, q = T, r = T
(a) p q

(b) p q
  
(c)  
p q r
    
(d)      
p r q r r p
     

80
CONDITIONAL PROPOSITION
Now let us learn about conditional propositions; firstly, its definition.
5.2
3. Let p and q be the propositions
p : The temperature is more than 40C.
q : It is raining.
Write the following propositions using p and q and logical
connectives.
(a) The temperature is more than 40C but it is raining.
(b) The temperature is less than 40C and it is raining.
(c) Either the temperature is more than 40C or it is raining.
4. Let p, q and r be the propositions
p : Ahmad is sick.
q : Ahmad misses his driving test.
r : Ahmad passes the test.
Express each of the following propositions as an English sentence:
(a) p q

(b) q r

(c) p q r
  
(d)    
p r q r
  
Create truth table for questions (b) and (c).
Definition 5.2a: If p and q are propositions, the compound proposition
if p then q
is called a conditional proposition and is denoted
p  q

The truth table for the conditional p  q is shown in Table 5.4.
Table 5.4: Conditional p  q
p q p  q
T
T
F
F
T
F
T
F
T
F
T
T
Let us look at two examples of conditional propositions.
Example 5.2a
If we define
p : The faculty is allocated another scholarship.
q : Ravie can further his studies at University of Sheffield.
The conditional statement p  q means
„If the faculty is allocated another scholarship, then Ravie can further his studies
at University of Sheffield.‰
The converse for p  q is q  p.
Example 5.2b
Let
p : 1 > 2,
q : 4 < 8.
Then p is false and q is true.
The proposition p  q, that is, „If 1 > 2 then 4 < 8‰ is true.
Therefore, the converse proposition q  p, that is „If 4 < 8 then 1 > 2‰ is false.

82
BICONDITIONAL PROPOSITION
What is a biconditional proposition?
The truth table for the biconditional proposition p  q is shown in Table 5.5.
Table 5.5: Proposition p  q
p q p  q
T
T
F
F
T
F
T
F
T
F
F
T
The biconditional proposition p  q is normally read as „p if and only if q‰ or „p
if q.‰ An alternative way to state this proposition is „p is a necessary and sufficient
condition for q.‰ Let us look at some examples of biconditional propositions.
Example 5.3a
If we define
p: 1 < 5, q: 2 < 8
then the statement
1 < 5 if and only if 2 < 8
can be written symbolically as
p  q
Since p and q are both true, the statement p  q is true.
An alternative way to state the statement above is: „A necessary and sufficient
condition for 1 < 5 is that 2 < 8.‰
5.3
Definition 5.3a: If p and q are propositions, the compound proposition
(p  q)  (q  p)
is called a biconditional proposition and is denoted as
p  q

Example 5.3b
If we define
p: Ammar can buy shoes online.
q: Ammar has a credit card.
Then the statement
Ammar can buy shoes online if and only if Ammar has a credit card
can be written symbolically as
p  q
An alternative way to state the statement above is: „A necessary and sufficient
condition for Ammar can buy shoes online is that Ammar has a credit card.‰
ACTIVITY 5.2
1. If each of the following statements define p and q, then write in
the form „if p, then q.‰
(a) It rains whenever the wind blows from the north-east.
(b) That the Pistons win the championship implies that they
beat the Lakers.
(c) It is necessary to walk a few kilometre to get to the top of the
mountain.
(d) To get accepted into OUM, you must have at least 3.0 CGPA.
(e) If you drive more than 110 km/h, you will get a ticket.
(f) I will remember to send you the address only if you send me
an e-mail message.
(g) To be a citizen of this country (Malaysia), it is sufficient that
you were born in this country.
(h) That you get the job implies that you had the best
credentials.

84
TAUTOLOGIES, CONTRADICTIONS AND
LOGICAL EQUIVALENCE
What do tautology and contradiction mean?
Example 5.4a
p p
  is an example of tautology.
p p
  is an example of contradiction.
How about logically equivalence propositions?
5.4
2. If each of the following statements define p and q, then write in
the form „p if and only if q.‰
(a) If it is hot outside you buy an ice cream cone, and if you
buy an ice cream cone it is hot outside.
(b) You get promoted only if you have connections, and you
have connections only if you get promoted.
(c) For you to pass this course, it is necessary and sufficient
that you learn how to solve most of the problems.
Definition 5.4a: A proposition p that is always true is called a tautology. A
proposition p that is always false is called a contradiction.
Definition 5.4b: Suppose that the compound proposition p is made up of
propositions p1⁄. pn and compound proposition q is made up of
propositions q1⁄. qn, we say that p and q are logically equivalent and write
p  q
provided that given any truth values of p1 ⁄. pn and truth values of q1 ⁄.
qn, either p and q are both true or p and q are both false.

Example 5.4b
Show that the negation of p q
 is logically equivalent to p q
  .
We must show that  
p q p q
    .
By writing the truth table for  
p q
  and p q
  , we can verify that given
any truth value of p and q, either p and q are both true or p and q are both false
(see Table 5.6).
Table 5.6:  
p q p q
   
p q q pq (pq) pq
T
T
F
F
T
F
T
F
F
T
F
T
T
F
T
T
F
T
F
F
F
T
F
F
Thus  
p q
  is logically equivalent to p q
  . We can write
 
p q p q
    .
Some of the properties of ,and negation is given in the following theorems.
Theorem 5.4a: The operations for propositions have the following properties
(see Table 5.7).
Table 5.7: Theorem 5.4a
Properties Propositions
(a) Commutative properties
p  q ≡q  p
p  q ≡ q  p
(b) Associative properties
p  (q  r) ≡ (p  q)  r
p  (q  r) ≡ (p  q)  r
(c) Distributive properties
p  (q  r) ≡ (p  q)  (p  r)
p  (q  r) ≡ (p  q)  (p  r)
(d) Idempotent properties
p  p ≡ p
p  p ≡ p
(e) Properties of negation  
p p
  

86
Some of the properties of conditional propositions are given in the following
theorem:
CONTRAPOSITIVE AND CONVERSE
Before we end this topic, let us learn about contrapositive and converse.
5.5
Theorem 5.4b (De MorganÊs Law):
1.      
p q p q
     
2.      
p q p q
     
Theorem 5.4c:
1.    
p q p q
   
2.    
p q p q
   
For each pair of the propositions below, state whether they are
logically equivalent.
(a) ,
p p q

(b) ,
p q p q
  
(c) ,
p q p q
 
(d)    
,
p p r p q r
    
(e)      
,
p q r p r q r
    
(f) ,
p q q p
   
(g) ,
p q p q
 
(h)    
,
P p q r Q p p r
     
ACTIVITY 5.3

Notice the difference between the contrapositive and the converse. The
converse of a conditional proposition merely reverses the roles of p and q
 
q p
 , whereas the contrapositive reverses the roles of p and q and
negates each of them. Let us look at an example.
Example 5.5a
If we define
p : 3 < 4,
q : 5 > 8,
Then the given proposition „if 3 < 4 then 5 > 8‰ may be written symbolically as
p  q
The converse is q  p, or in words „if 5 > 8, then 3 < 4.‰
The contrapositive is q p
   , or in words „if 5 is not greater than 8, then 3 is
not less than 4‰ or „if 5 < 8, then 3 > 4.‰
We see that p  q is false, then q p
   is false.
An important fact is that a conditional proposition and its contrapositive are
logically equivalent. This can be shown by using the truth table as shown in
Table 5.9.
Table 5.8: p q q p
    
p q q pq (pq) pq
T T F F T T
T F F T F F
F T T F T T
F F T T T T
Definition 5.5a: The contrapositive for a conditional proposition p  q is the
proposition q p
  

88
• Proposition is a statement that is either true or false.
• The propositions in statements can be formulated into symbolic expressions.
• The compound propositions can be proved by using truth tables and laws of
logic.
• The operations on compound propostions are tautologies, contradiction,
logical equivalence, contrapositive and converse.
Biconditional proposition
Conditional proposition
Conjunction
Contradiction
Contrapositive
Converse
Disjunction
Logical equivalence
Negation
Proposition
Tautologies
If each of the following statements define p and q, then state the
converse and contrapositive in an English sentence.
(a) I go to town whenever there is a need to buy a new cloth.
(b) A positive integer is a prime only if it has no division other than 1
and itself.
(c) If it rains tonight, then I will stay at home.
(d) I go to the beach whenever it is a holiday.
ACTIVITY 5.4

Pearson.
Logic. (2009). Retrieved from http://www.cs.odu.edu/~toida/nerzic/content
/logic/intr_to_logic.html
Logic. (2015). Retrieved from http://en.wikibooks.org/wiki/Discrete_
Mathematics/Logic
Poly, G. (1945). How to solve it. Princeton: Princeton University Press.
McGraw Hill.
Suber, P. (1997). Propositional Logic Terms and Symbols. Retrieved from
http://www.earlham.edu/~peters/courses/log/terms2. htm

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