Relation and Function

RELATIONS AND FUNCTIONS

INTRODUCTION
We have studied in class XI about a relation, its domain and range. There we have also
studied representation of a relation in different forms and number of relations from one set to
another set and number of relations on a set. There we have also seen that a function is a
special type of relation. Here we shall study different types of relations on a set. We recall
that a relation on a set A is a subset of A × A.  and A × A are subsets of A × A and so they
are also relations on A. These two extreme relations on A have been given special names,
empty relation and universal relation respectively. Here, we shall also study other relations on
A.

TYPES OF RELATIONS
TRIVIAL RELATIONS
Trivial relations are of two types:

o Empty relation
o Universal relation
A relation in a set A is called an empty relation if no element of A is related to any element of
A, i.e., R = ⊂ A × A.

For example: Consider a relation R in set A = {2, 4, 6} defined by R = {(a, b): a + b is odd,
where a, b ∈ A}. The relation R is an empty relation since for any pair (a, b) ∈ A × A, a + b is
always even.

A relation R in a set A is called a universal relation if each element of A is related to every
element of A, i.e., R = A × A.
For example: Let A be the set of all students of class XI. Let R be a relation in set A defined
by R = {(a, b): the sum of the ages of a and b is greater than 10 years}

The relation R is a universal relation because it is obvious that the sum of the ages of two
students of class XI is always greater than 10 years.

EQUIVALENCE RELATION
One of the most important relations, which plays a significant role in Mathematics, is an
equivalence relation. To study equivalence relation, we first consider three types of relations,
namely reflexive, symmetric and transitive.

A relation R in a set A is called reflexive if (a, a) ∈ R for every a ∈ A.


For example: A relation R in set A   0,  defined by R = {sin a = sin b; a, b ∈ A} is a
 2
reflexive relation since sin a = sin a  a  A.

, A relation R in a set A is called symmetric if (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1,
a2∈ A.


For example: A relation in the set A   0,  defined by R = {sin a = sin b; a, b ∈ A} is a
 2
symmetric relation. Since for a, b ∈ A, sin a = sin b implies sin b = sin a. So, (a, b) ∈ R ⇒ (b,
a) ∈ R.

A relation R in a set A is called transitive if (a1, a2) ∈ R, and (a2, a3) ∈ R together imply that
(a1, a3) ∈ R, for all a1, a2, a3∈ A.


For example: A relation in the set A   0,  defined by R = {sin a = sin b, a, b ∈ A} is a
 2
transitive relation. Since for a, b, c ∈ A, let (a, b), (b, c) ∈ R.

 sin a = sin b and sin b = sin c

 sin a = sin c

 (a, c) ∈ R

A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and
transitive.

For example: Relation R in the set A   0,  defined by R = {sin a = sin b; a, b ∈ A} is an
 2
equivalence relation.

Equivalence Classes
Every arbitrary equivalence relation R in a set X divides X into mutually disjoint subsets (Ai)
called partitions or subdivisions of X satisfying the following conditions:
All elements of Ai are related to each other for all i
No element of Ai is related to any element of Aj whenever i ≠ j

Ai ∪Aj = X and Ai ∩ Aj= Φ, i ≠ j
These subsets (Ai) are called equivalence classes.

For an equivalence relation in a set X, the equivalence class containing a ∈ X, denoted by [a],
is the subset of X containing all elements b related to a.

TYPES OF FUNCTIONS
The notion of a function along with some special functions like identity function, constant
function, polynomial function, rational function, modulus function, signum function etc.
along with their graphs have been given in Class XI. Addition, subtraction, multiplication and
division of two functions have also been studied. As the concept of function is of paramount
importance in mathematics and among other disciplines as well, we would like to extend our