NCERT Exemplar Solutions
Class 12 - Mathematics - Chapter 1: RELATIONS AND FUNCTIONS

Long Answer Questions

Given \(A = \{1,2,3,4\}\), define relations on \(A\) having the properties:

• (a) reflexive, transitive but not symmetric
• (b) symmetric but neither reflexive nor transitive
• (c) reflexive, symmetric and transitive

Let \(R\) be a relation defined on the set of natural numbers \(\mathbb{N}\) as follows:

\[ R = \{(x,y) : x,y \in \mathbb{N},\ 2x + y = 41 \}. \]

Find the domain and range of the relation \(R\). Also verify whether \(R\) is reflexive, symmetric and transitive.

Given \(A = \{2,3,4\}\) and \(B = \{2,5,6,7\}\), construct an example of each of the following:

• (a) an injective mapping from \(A\) to \(B\)
• (b) a mapping from \(A\) to \(B\) which is not injective
• (c) a mapping from \(B\) to \(A\)

Give an example of a map:

• (i) which is one-one but not onto
• (ii) which is not one-one but onto
• (iii) which is neither one-one nor onto

Let \(A = \mathbb{R} - \{3\}\) and \(B = \mathbb{R} - \{1\}\). Let \(f : A \to B\) be defined by

\[ f(x) = \dfrac{x - 2}{x - 3}. \]

Show that \(f\) is bijective.

Let \(A = [-1,1]\). Discuss whether the following functions defined on \(A\) are one-one, onto or bijective:

(i) \(f(x) = \dfrac{x}{2}\)
(ii) \(g(x)=|x|\)
(iii) \(h(x) = x|x|\)
(iv) \(k(x)=x^2\)

Each of the following defines a relation on \(\mathbb{N}\):

(i) \(x\) is greater than \(y\)
(ii) \(x + y = 10\)

Determine which of the above relations are reflexive, symmetric and transitive.

Let \(A = \{1,2,3,\ldots,9\}\) and \(R\) be the relation defined on \(A \times A\) by

\[(a,b) R (c,d) \text{ if } a+d = b+c.\]

Prove that \(R\) is an equivalence relation and find the equivalence class \([(2,5)]\).

Using the definition, prove that the function \(f : A \to B\) is invertible if and only if \(f\) is both one-one and onto.

Functions \(f,g : \mathbb{R} \to \mathbb{R}\) are defined by \(f(x)=x^2+3x+1\) and \(g(x)=2x−3\). Find:

(i) \(f \circ g\)
(ii) \(g \circ f\)
(iii) \(f \circ f\)
(iv) \(g \circ g\)

Let * be the binary operation defined on \(\mathbb{Q}\). Determine which of the following operations are commutative:

(i) \(a * b = a - b\)
(ii) \(a * b = a^2 + b^2\)
(iii) \(a * b = a + ab\)
(iv) \(a * b = (a - b)^2\)

Let * be a binary operation defined on \(\mathbb{R}\) by \(a * b = 1 + ab\). Then determine whether the operation is:

(i) commutative but not associative
(ii) associative but not commutative
(iii) neither commutative nor associative
(iv) both commutative and associative