NCERT Exemplar Solutions
Class 12 - Mathematics

Chapter 1: RELATIONS AND FUNCTIONS

Let \(A = \{a, b, c\}\) and the relation \(R\) be defined on \(A\) as follows: \(R = \{(a,a),(b,c),(a,b)\}\). Then, write minimum number of ordered pairs to be added in \(R\) to make \(R\) reflexive and transitive.

Let \(D\) be the domain of the real valued function \(f\) defined by \(f(x) = \sqrt{25 - x^2}\). Then write \(D\).

Let \(f, g : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = 2x + 1\) and \(g(x) = x^2 - 2\). Find \(g \circ f\).

Let \(f : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = 2x - 3\). Write \(f^{-1}\).

If \(A = \{a,b,c,d\}\) and the function \(f = \{(a,b),(b,d),(c,a),(d,c)\}\), write \(f^{-1}\).

If \(f : \mathbb{R} \to \mathbb{R}\) is defined by \(f(x) = x^2 - 3x + 2\), write \(f(f(x))\).

Is \(g = \{(1,1),(2,3),(3,5),(4,7)\}\) a function? If \(g(x) = \alpha x + \beta\), then what are the values of \(\alpha\) and \(\beta\)?

Are the following sets of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
(i) \(\{(x,y): x \text{ is a person}, y \text{ is the mother of } x\}\)
(ii) \(\{(a,b): a \text{ is a person}, b \text{ is an ancestor of } a\}\)

If mappings \(f\) and \(g\) are given by \(f = \{(1,2),(3,5),(4,1)\}\) and \(g = \{(2,3),(5,1),(1,3)\}\), write \(g \circ f\).

Let \(C\) be the set of complex numbers. Prove that the mapping \(f:C \to \mathbb{R}\) defined by \(f(z)=|z|\) is neither one-one nor onto.

Let the function \(f:\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \cos x\). Show that \(f\) is neither one-one nor onto.

Let \(X = \{1,2,3\}\) and \(Y = \{4,5\}\). Determine whether the following sets of \(X \times Y\) are functions from \(X\) to \(Y\):
(i) \(f = \{(1,4),(1,5),(2,4),(3,5)\}\)
(ii) \(g = \{(1,4),(2,4),(3,4)\}\)
(iii) \(h = \{(1,4),(2,5),(3,5)\}\)
(iv) \(k = \{(1,4),(2,5)\}\)

If functions \(f:A \to B\) and \(g:B \to A\) satisfy \(g \circ f = I_A\), show that \(f\) is one-one and \(g\) is onto.

Let \(f:\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \dfrac{1}{2 - \cos x}\). Find the range of \(f\).

Let \(n\) be a fixed positive integer. Define a relation \(R\) in \(\mathbb{Z}\) as follows: for \(a,b \in \mathbb{Z}\), \(aRb\) iff \(a-b\) is divisible by \(n\). Show that \(R\) is an equivalence relation.

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

Let the relation \(R\) be defined in \(\mathbb{N}\) by \(aRb\) if \(2a + 3b = 30\). Then \(R = \) ____.

Let the relation \(R\) be defined on the set \(A=\{1,2,3,4,5\}\) by \(R=\{(a,b):|a^2-b^2|<8\}\). Then \(R\) is given by ____.

Let \(f=\{(1,2),(3,5),(4,1)\}\) and \(g=\{(2,3),(5,1),(1,3)\}\). Then \(g\circ f =\) ____ and \(f\circ g=\) ____.

Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=\dfrac{x}{\sqrt{1+x^2}}\). Then \((f\circ f\circ f)(x)=\) ____.

If \(f(x)=4-(x-7)^3\), then \(f^{-1}(x)=\) ____.

Let R = {(3,1), (1,3), (3,3)} be a relation defined on the set A = {1,2,3}. Then R is symmetric, transitive but not reflexive.

Let f : \(\mathbb{R} \to \mathbb{R}\) be the function defined by f(x) = sin(3x + 2) for all x ∈ \(\mathbb{R}\). Then f is invertible.

Every relation which is symmetric and transitive is also reflexive.

An integer m is said to be related to another integer n if m is an integral multiple of n. This relation in \(\mathbb{Z}\) is reflexive, symmetric and transitive.

Let A = {0, 1} and N be the set of natural numbers. Then the mapping f : N → A defined by f(2n − 1) = 0 and f(2n) = 1 for all n ∈ N is onto.

The relation R on the set A = {1, 2, 3} defined as R = {(1,1), (1,2), (2,1), (3,3)} is reflexive, symmetric and transitive.

The composition of functions is commutative.

The composition of functions is associative.

Every function is invertible.

A binary operation on a set has always the identity element.