NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 1 - Relations And Functions [Latest edition]

Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?

For the set A = {1, 2, 3}, define a relation R in the set A as follows: R = {(1, 1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

Let R be the equivalence relation in the set Z of integers given by R = {(a, b): 2 divides a – b}. Write the equivalence class [0]

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g

Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f^–1 

Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?

If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g

If A = {1, 2, 3} and f, g are relations corresponding to the subset of A × A indicated against them, which of f, g is a function? Why?
f = {(1, 3), (2, 3), (3, 2)}
g = {(1, 2), (1, 3), (3, 1)}

If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f^–1

In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?

In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R

Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto

Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f

Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f^–1.

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b for a, b ∈ Q

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = `"ab"/4` for a, b ∈ Q.

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b + ab for a, b ∈ Q

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = ab² for a, b ∈ Q

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.

Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.

Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.

Let f: R → R be defined by f(x) = sin x and g: R → R be defined by g(x) = x 2 , then f o g is ______.

Let f: R → R be defined by f(x) = 3x – 4. Then f^–1(x) is given by ______.

Let f: R → R be defined by f(x) = x² + 1. Then, pre-images of 17 and – 3, respectively, are ______.

For real numbers x and y, define xRy if and only if x – y + `sqrt(2)` is an irrational number. Then the relation R is ______.

Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______

The domain of the function f: R → R defined by f(x) = `sqrt(x^2 - 3x + 2)` is ______

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______

Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets

Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.

Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.

Let A be a finite set. Then, each injective function from A into itself is not surjective.

For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.

For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.

Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.

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: R → R be defined by f(x) = 2x + 1 and g(x) = x² – 2, ∀ x ∈ R, respectively. Then, find gof

Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. 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: R → R is defined by f(x) = x² – 3x + 2, write f(f (x))

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g (x) = αx + β, then what value should be assigned to α and β

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}

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

Let C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.

Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

f = {(1, 4), (1, 5), (2, 4), (3, 5)}

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

g = {(1, 4), (2, 4), (3, 4)}

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

h = {(1,4), (2, 5), (3, 5)}

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

k = {(1,4), (2, 5)}

If functions f: A → B and g: B → A satisfy gof = I_A, then show that f is one-one and g is onto

Let f: R → R be the function defined by f(x) = `1/(2 - cosx)` ∀ x ∈ R.Then, find the range of f

Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric

If A = {1, 2, 3, 4 }, define relations on A which have properties of being: 
symmetric but neither reflexive nor transitive

If A = {1, 2, 3, 4 }, define relations on A which have properties of being: 
reflexive, symmetric and transitive

Let R be relation defined on the set of natural number N as follows:
R = {(x, y): x ∈N, y ∈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}, B = {2, 5, 6, 7}. Construct an example of the following:
an injective mapping from A to B

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from A to B which is not injective

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from B to A

Give an example of a map which is one-one but not onto

Give an example of a map which is not one-one but onto

Give an example of a map which is neither one-one nor onto

Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

f(x) = `x/2`

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

g(x) = |x|

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

h(x) = x|x|

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

k(x) = x² 

The following defines a relation on N:
x is greater than y, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:
x y is square of an integer x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.

Let A = {1, 2, 3, ... 9} and R be the relation in A × A defined by (a, b) R(c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]

Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find f o g

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find g o f

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find f o f

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find g o g

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a – b ∀ a, b ∈ Q

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a² + b² ∀ a, b ∈ Q

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a + ab ∀ a, b ∈ Q

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = (a – b)² ∀ a, b ∈ Q

Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.

Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is ______.

The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.

Let us define a relation R in R as aRb if a ≥ b. Then R is ______.

Let A = {1, 2, 3} and consider the relation R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R is ______.

The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.

Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.

Let f: R → R be defined by f(x) = 3x 2 – 5 and g: R → R by g(x) = `x/(x^2 + 1)` Then gof is ______.

Which of the following functions from Z into Z are bijections?

Let f: R → R be the functions defined by f(x) = x³ + 5. Then f^–1(x) is ______.

Let f: A → B and g: B → C be the bijective functions. Then (g o f)^–1 is ______.

Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.

Let f: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",",  "if"  x  "is rational"),(1 - x",",  "if"  x  "is irrational"):}`. Then (f o f) x is ______.

Let f: `[2, oo)` → R be the function defined by f(x) = x² – 4x + 5, then the range of f is ______.

Let f: N → R be the function defined by f(x) = `(2x - 1)/2` and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) `3/2` is ______.

Let f: R → R be defined by f(x) = `{{:(2x",", x > 3),(x^2",", 1 < x ≤ 3),(3x",", x ≤ 1):}`. Then f(–1) + f(2) + f(4) is ______.

Let f: R → R be given by f(x) = tan x. Then f^–1(1) is ______.

Let the relation R be defined in 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) : |a² – b²| < 8. Then R is given by ______.

Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.

Let f: R → R be defined by f(x) = `x/sqrt(1 + x^2)`. Then (f o f o 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: R → R be the function defined by f(x) = sin (3x+2) ∀ x ∈ 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 a integral multiple of n. This relation in 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, f(2n) = 1, ∀ 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.