Chapter 2: Mathematical Preliminaries ~ 45
EXAMPLE 2.10
If R = {(a, b), (b, c), (c, a)} is a relation in {a. b. c}, find R*.
Solution
From Example 2.9,
R* = R+ u {(a, a), (b, b), (c, c)}
= {(a, b), (b, c), (c. a). (a, c), (b, a), (c, b), (a. a), (b, b), (c, c)}
EXAMPLE 2.11
What is the symmetric closure of relation R in a set S?
Solution
Symmetric closure of R = R u {(b, a) IaRb}.
2.1.6 FUNCTIONS
The concept of a function arises when we want to associate a unique value (or
result) with a given argument (or input).
Definition 2.7 A function or map f from a set X to a set Y is a rule which
associates to every element x in X a unique element in Y. which is denoted by
j(x). The element f(x) is called the image of .y under f The function is denoted
by f X ~ Y.
Functions can be defined either (i) by giving the images of all elements
of X, or (ii) by a computational rule which computes f(x) once x is given.
EXAMPLES (a)f: {l. 2. 3. 4} ~ {a, b, c} can be defined byf(1) = a,
f(2) = c, f(3) = a, f(4) = b.
(b) f: R ~ R can be defined by f(x) = .J + 2x + 1 for every x in R.
(R denotes the set of all real numbers.)
Definition 2.8 f: X ~ Y is said to be one-to-one (or injective) if different
elements in X have different images. i.e. f(Xl) =i = f(X2) when Xl =i= X2'
Note: To prove that f is one-to-one. we prove the following: Assume
f(.I:J = f(X2) and show that Xl = X2'
Definition 2.9 f: X ~ Y is onto (sUljective) if every element y in Y is the
image of some element x in X.
Dpfmition 2.10 f: X ~ Y is said to be a one-to-one correspondence or
biJection if f is both one-to-one and onto.