1
Discrete Theory, Relations
and Functions
21.1 INTRODUCTION
The major objective of the study of this subject is to study the theory of discrete objects and
relationship among them. The term discrete objects refers a variety of items that we frequently
seen and go through in our day today life such as students, books, programs, numbers, projects
etc. We study some of the basic concepts dealing with different kinds of discrete objects under
the theory of set algebra which we discussed next. Initially, the notation of set theory is intro-
duced and certain operations are defined. The concepts of relations and functions are pre-
sented after a discussion of algebra of sets. We conclude this chapter with the study of natural
numbers, Peano axioms and mathematical induction.1.2 ELEMENTARY THEORY OF SETS
In this section we start our study to introduce the notation used for specifying sets. A set is the
known collection of objects that are distinct in nature.
For example,
l A set of books,
l A set of alphabets,
l A set of real numbers,
l A set of Ist semester of MCA students,
l A group of students of UP Technical University, Lucknow,
l A group of meritorious students of the university,
l A collection of coins,
l A aggregation of live projects,
The words ‘group’, ‘collection’, ‘aggregation’, are have similar meaning of set. The set is
recognizes by the uniqueness of its members. The important characteristic of the set is that its
members share some common, unique, and well defined property through which the set is
recognized or named. A set is represented by a symbol. We use the convention capital letter to
represent a set, and lower case letters to represent the members of the set. For example X is
the set of alphabets i.e. X = {a, b, c, ...., z}; the set of natural numbers N = {0, 1, 2, ....} etc. The
objects of the set are often called the members of the set or element of the set for example, a, b,
c, ..., z are the members of the set X. It is discretionary that in the set the members occur in
any order.
Let set X = {x, y, z} then elements x, y, and z are the members of the set X, this can also
be represented by x ∈ X, y ∈ X and z ∈ X but for the element a ∉ X it means that a is not the