Relations and Functions

Previous Years' CBSE Board Questions

1.2 Types of Relations

MCQ

1 Let <math>A = \{3, 5\}</math>. Then number of reflexive relations on A is

1. 2 (b) 4
2. 0 (d) 8

2 Let R be a relation in the set N given by <math>R = \{(a, b) : a = b - 2, b > 6\}</math>. Then

1. (8, 7) ∈ R (b) (6, 8) ∈ R
2. (3,8) ∈ R (d) (2, 4) ∈ R

1. A relation R is defined on N. Which of the following is the reflexive relation?
2. R = {(x, y): x > y, x, y ∈ N}
3. R = {(x, y): x + y = 10, x, y ∈ N}
4. R = {(x, y): xy is the square number, x, y ∈ N}
5. R = {(x, y): x + 4y = 10, x, y ∈ N}

(Term I, 2021-22) An

1. The number of equivalence relations in the set (1, 2, 3) containing the elements (1, 2) and (2, 1) is
2. 0 (b) 1
3. 2 (d) 3 (Term I, 2021-22)

1. A relation R is defined on Z as aRb if and only if <math>a^2 - 7ab + 6b^2 = 0</math>. Then, R is
2. reflexive and symmetric
3. symmetric but not reflexive
4. transitive but not reflexive
5. reflexive but not symmetric

(Term I, 2021-22) [Apr

1. Let A = {1, 3, 5}. Then the number of equivalence relations in A containing (1, 3) is
2. 1 (b) 2
3. 3 (d) 4

1. The relation R in the set <math>\{1, 2, 3\}</math> given by <math>R = \{(1, 2), 2\}</math> (2, 1), (1, 1)} is
2. symmetric and transitive, but not reflexive
3. reflexive and symmetric, but not transitive
4. symmetric, but neither reflexive nor transitive
5. an equivalence relation

V5A (1 mark)

Write the smallest reflexive relation on set <math>A = \{a, b, c\}</math>.

(2021 C)

A relation R in a set A is called _____, if <math>(a_1, a_2) \in R</math> implies <math>(a_2, a_1) \in R</math>, for all <math>a_1, a_2 \in A</math>. (2020)

A relation in a set A is called _ relation, if each element of A is related to itself. (2020) R

1. If R = {(x, y) : x + 2y = 8} is a relation on N, write the range of R. (Al 2014)

Let R = {(a, a³) : a is a prime number less than 5} be a

relation. Find the range of R. (Foreign 2014)

13. Let R be the equivalence relation in the set

<math>A = \{0, 1, 2, 3, 4, 5\}</math> given by <math>R = \{(a, b) : 2 \text{ divides } (a - b)\}.</math>

Write the equivalence class [0]. (Delhi 2014 C)

(2 marks)

Check if the relation R in the set R of real numbers

defined as <math>R = \{(a, b) : a < b\}</math> is (i) symmetric,

1. transitive. (2020)

15. Let W denote the set of words in the English

dictionary. Define the relation R by

<math>R = \{(x, y) \in W \times W \text{ such that } x \text{ and } y \text{ have at least one } x \text{ and } y \text{ have at least one } y \text{ have at least one } y \text{ have at least one } y \text{ have at least one } y \text{ have at least one } y \text{ have at </math>

letter in common). Show that this relation R is reflexive and symmetric,

but not transitive. (2020)

LAI (4 marks)

1. Show that the relation R in the set <math>A = \{1, 2, 3, 4, 5, 6\}</math> given by <math>R = \{(a, b) : |a - b| \text{ is divisible by 2}\}</math> is an equivalence relation. (2020)
2. Check whether the relation R defined on the set <math>A = \{1, 2, 3, 4, 5, 6\}</math> as <math>R = \{(a, b) : b = a + 1\}</math> is reflexive, symmetric or transitive. (2019)
3. Show that the relation R on the set Z of all integers, given by <math>R = \{(a, b) : 2 \text{ divides } (a - b)\} \text{ is an equivalence relation.}</math> (2019)
4. Show that the relation R on <math>\mathbb{R}</math> defined as <math>R = \{(a,b): a \le b\}</math>, is reflexive and transitive but not symmetric. (NCERT, Delhi 2019)

Show that the relation S in the set A={x∈Z:0≤x≤12}

given by <math>S = \{(a, b) : a, b \in \mathbb{Z}, |a - b| \text{ is divisible by 3}\}</math> is

an equivalence relation. (Al 2019) [Ap]

21. Let A = (1, 2, 3, ..., 9) and R be the relation in

<math>A \times A</math> defined by <math>(a, b) R (c, d)</math> if <math>a + d = b + c</math>

for <math>(a, b)</math>, <math>(c, d)</math> in <math>A \times A</math>. Prove that R is an equivalence

relation. Also obtain the equivalence class [(2, 5)].

(Delhi 2014)

22. Let R be a relation defined on the set of natural

numbers N as follow:

<math>R = \{(x, y) \mid x \in N, y \in N \text{ and } 2x + y = 24\}</math>

Find the domain and range of the relation R.

Also, find if R is an equivalence relation or not.

(Delhi 2014 C) [An

LAII (5/6 marks)

23. If N denotes the set of all natural numbers and R

is the relation on <math>N \times N</math> defined by <math>(a, b) R</math> <math>(c, d)</math>, if

<math>ad(b + c) = bc(a + d)</math>. Show that R is an equivalence

relation. (2023, Delhi 2015)

Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a - b| is divisible by 4], is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]. (2018)

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by <math>R = \{(a, b) : |a - b| \text{ is divisible by 2}\}</math> is an equivalence relation. Write all the equivalence classes of R.

(AI 2015 C)

1.3 Types of Functions

MCQ

1. The function f: R → R defined by f(x) = 4 + 3 cosx is
2. bijective (b) one-one but not onto
3. onto but not one-one
4. neither one-one nor onto (Term I, 2021-22) [An]

The number of functions defined from <math>\{1, 2, 3, 4, 5\} \rightarrow \{a, b\}</math> which are one-one is

1. 5 (b) 3
2. 2 (d) 0 (Term I, 2021-22)

1. Let f: R → R be defined by f(x) = 1/x, for all x ∈ R, Then, fis
2. one-one (b) onto
3. bijective (d) not defined (Term I, 2021-22)

OR (2023) An

1. The function f: N → N is defined by <math>\frac{n+1}{2}</math>, if n is odd <math>f(n) =</math> n 2 if n is even The function f is
2. bijective
3. one-one but not onto
4. onto but not one-one
5. neither one-one nor onto

(Term I, 2021-22) [[v]

VSA (1 mark)

1. If f = {(1, 2), (2, 4), (3, 1), (4, k)} is a one-one function from set A to A, where <math>A = \{1, 2, 3, 4\}</math>, then find the

value of k. (2021 C)

31. Case Study: An organization conducted bike race

under two different categories - Boys and girls.

There were 28 participants in all. Among all of them,

finally three from category 1 and two from category

2 were selected for the final race. Ravi forms two

sets B and G with these participants for his college

project. Let <math>B = \{b_1, b_2, b_3\}</math> and <math>G = \{g_1, g_2\}</math>, where B represents

the set of Boys selected and G the set of Girls

selected for the final race.

LAI (4 marks)

Based on the above information, answer the

following questions.

1. (ii) Among all the possible relations from B to G, how many functions can be formed from B to G? (iii) Let R: B → B be defined by R = {(x, y) : x and y are students of the same sex). Check if R is an equivalence relation.

A function <math>f: B \rightarrow G</math> be defined by <math>f = \{(b_1, g_1), (b_2, g_2),</math>

(b3, g1). Check if f is bijective, justify your answer.

1. Let <math>f: \mathbb{R} - \left\{ -\frac{4}{3} \right\} \to \mathbb{R}</math> be a function defined as <math>f(x) = \frac{4x}{2x+4}</math>. Show that f is a one-one function. Also, check whether f is an onto function or not. (2023)
2. Show that the function f: (-∞, 0) → (-1, 0) defined by <math display="block">f(x) = \frac{x}{1+|x|}, x \in (-\infty, 0) \text{ is one-one and onto.} </math> (2020)

CBSE Sample Questions

1.2 Types of Relations

MCQ

1. A relation R in set <math>A = \{1, 2, 3\}</math> is defined as <math>R = \{(1, 1), (1, 2), (2, 2), (3, 3)\}</math>. Which of the follow- ing ordered pair in R shall be removed to make it an equivalence relation in A?
2. (1, 1) (b) (1, 2) (c) (2, 2) (d) (3, 3)

Let the relation R in the set <math>A = \{x \in Z : 0 \le x \le 12\}</math>,

given by <math>R = \{(a, b) : |a - b| \text{ is a multiple of 4.}\}</math> Then [1],

the equivalence class containing 1, is

1. [1, 5, 9] (b) {0, 1, 2, 5}
2. 0 (d) A (Term I, 2021-22) Ev

V5A (1 mark)

How many reflexive relations are possible in a set A

whose <math>n(A) = 3</math>? (2020-21) Ap

(Term I, 2021-22) An