NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 1

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Overview

1.1.1 Relation

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product . The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range is always a subset of codomain.

1.1.2 Types of Relations

A relation R in a set A is subset of . Thus empty set φ and are two extreme relations.

(i) 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., .

(ii) 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.,

(iii) A relation R in A is said to be reflexive if for all , R is symmetric if and it is said to be transitive if and . Any relation which is reflexive, symmetric and transitive is called an equivalence relation.

Note: An important property of an equivalence relation is that it divides the set into pairwise disjoint subsets called equivalent classes whose collection is called a partition of the set. Note that the union of all equivalence classes gives the whole set.

1.1.3 Types of Functions

(i) A function is defined to be one-one (or injective), if the images of distinct elements of under are distinct, i.e.,

.

(ii) 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 there exists an element such that .

(iii) A function is said to be one-one and onto (or bijective), if is both one-one and onto.

1.1.4 Composition of Functions

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

g o f (x) = g (f (x)), ∀ x ∈ A.

(ii) If and are one-one, then o is also one-one

(iii) If and are onto, then g o is also onto.

However, converse of above stated results (ii) and (iii) need not be true. Moreover, we have the following results in this direction.

(iv) Let and be the given functions such that is one-one. Then is one-one.

(v) Let and be the given functions such that is onto. Then is onto.

1.1.5 Invertible Function

(i) A function is defined to be invertible, if there exists a function such that and . The function g is called the inverse of f and is denoted by .

(ii) A function is invertible if and only if is a objective function.

(iii) If , and are functions, then .

(iv) Let and be two invertible functions. Then is also invertible with .

1.1.6 Binary Operations

(i) A binary operation on a set A is a function . We denote by.

(ii) A binary operation on the set is called commutative, if for every.

(iii) A binary operation is said to be associative if , for every .

(iv) Given a binary operation, an element , if it exists, is called identity for the operation , if .

(v) Given a binary operation : , with the identity element in , an element , is said to be invertible with respect to the operation , if there exists an element in A such that and is called the inverse of and is denoted by .

1.2 Solved Examples

Short Answer (S.A.)

Question 1:

Let and define a relation R on A as follows:

.

Is R reflexive? Symmetric? Transitive?

Answer:

R is reflexive and symmetric, but not transitive since for and whereas .

Question 2:

For the set , define a relation R in the set A as follows:

.

Write the ordered pairs to be added to R to make it the smallest equivalence relation.

Answer:

is the single ordered pair which needs to be added to R to make it the smallest equivalence relation.

Question 3:

Let R be the equivalence relation in the set of integers given by . Write the equivalence class .

Answer: