RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS

Class 12, NCERT Chapter 1, Theory

1. Introduction

The concept of the term ‘relation’ in mathematics has been drawn from the meaning of the relation in the English language, according to which two objects or quantities are related if there is a recognizable connection or link between the two objects or quantities

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 recognizable connection or link between a and b.

 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): a – b = 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 | a – b | ≥ 0. 

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.

Definition 3 

A relation R in a set A is called 

 (i) reflexive, if (a, a) ∈ R, for every a ∈ A,

  (ii) symmetric, if (a1 , a2 ) ∈ R implies that (a2 , a1 ) ∈ R, for all a1 , a2 ∈ A.

   (iii) transitive, if (a1 , a2 ) ∈ R and (a2 , a3 ) ∈ R implies that (a1 , a3 )∈ R, for all a1 , a2 , a3 ∈ A.

Definition 4

 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric, and transitive. 

3.Types of Functions

Consider the functions f 1 , f 2 , f 3 and f 4 given by the following diagram

Figure1

Figure2

Figure3

Figure4

Definition 5 

A function f: X → Y is defined to be one-one (or injective) if the images of distinct elements of X under f are distinct, i.e., for every x1 , x2 ∈ X, f(x1 ) = f(x2 ) implies x1 = x2 . Otherwise, f is called many-one.

The function f 1 and f 4 in Fig (1) and (4) are one-one and the function f 2 and f 3 in Fig 1.2 (2) and (3) are many-one.

Definition 6

 A function f: X → Y is said to be onto (or surjective) if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f(x) = y. The function f 3 and f 4 in Fig(3),(4) are onto and the function f 1 in Fig  (1) is not onto as elements e, f in X2 are not the image of any element in X1 under f 1 .

Definition 7

 A function f: X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto. The function f 4 in Fig (4) is one-one and onto.

4.Composition of Functions and Invertible Function

In this section, we will study the composition of functions and the inverse of a bijective function. Consider the set A of all students, who appeared in Class X of a Board Examination in 2006. Each student appearing in the Board Examination is assigned a roll number by the Board which is written by the students in the answer script at the time of examination. In order to have confidentiality, the Board arranges to deface the roll numbers of students in the answer scripts and assigns a fake code number to each roll number. Let B ⊂ N be the set of all roll numbers and C ⊂ N be the set of all code numbers. This gives rise to two functions f: A→ B and g: B → C given by f(a) = the roll number assigned to the student a and g(b) = the code number assigned to the roll number b. In this process, each student is assigned a roll number through the function f and each roll number is assigned a code number through the function g. Thus, by the combination of these two functions, each student is eventually attached a code number. 

This leads to the following definition:

 Definition 8

 Let f: A → B and g: B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof: A → C given by

 gof (x) = g(f (x)), ∀ x ∈ A. 

Figure5

Definition 9

 A function f: X → Y is defined to be invertible if there exists a function g: Y → X such that gof = IX and fog = IY . The function g is called the inverse of f and is denoted by f –1 . Thus, if f is invertible, then f must be one-one and onto and conversely if f is one-one and onto, then f must be invertible. This fact significantly helps for proving a function f to be invertible by showing that f is one-one and onto, especially when the actual inverse of f is not to be determined.

Theorem 1

 If f : X → Y, g : Y → Z and h : Z → S are functions, then

 ho(gof) = (hog) o f.

 Proof

 We have

 ho(gof) (x) = h(gof(x)) = h(g(f(x))), ∀ x in X

 and (hog) of (x) = hog(f (x)) = h(g(f(x))), ∀ x in X.

 Hence,                                              ho(gof) = (hog) o f.