### Chapter 1 : RELATIONS AND FUNCTIONS

There is no permanent place in the world for ugly mathematics ... . It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognising one when we read it.
- G. H. HARDY

1.1 Introduction

Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are

(i)      {(a, b) A × B: a is brother of b},

(ii)     {(a, b) A × B: a is sister of b},

(1805-1859)

(iii)    {(a, b) A × B: age of a is greater than age of b},

(iv)     {(a, b) A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination},

(v)     {(a, b) A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B.

If (a, b) R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations.

In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations.

1.2 Types of Relations

In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set φ and A × A are two extreme relations. For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): ab = 10}. This is the empty set, as no pair (a, b) satisfies the condition a b = 10. Similarly, R = {(a, b) : | a b | 0} is the whole set A × A, as all pairs (a, b) in A × A satisfy | ab | 0. These two extreme examples lead us to the following definitions.

Definition 1 A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ ⊂ A × A.

Definition 2 A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A.

Both the empty relation and the universal relation are some times called trivial relations.

Empty Relation

Universal Relation

Example 1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = φ, showing that R is the empty relation. It is also obvious that the difference between heights of any two students of the school has to be less than 3 meters. This shows that R = A × A is the universal relation.

Remark In Class XI, we have seen two ways of representing a relation, namely roaster method and set builder method. However, a relation R in the set {1, 2, 3, 4} defined by R = {(a, b) : b = a + 1} is also expressed as a R b if and only if b = a + 1 by many authors. We may also use this notation, as and when convenient.

If (a, b) R, we say that a is related to b and we denote it as a R b.

One of the most important relation, 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.