CBSE (Commerce) Class 11CBSE
Share

Books Shortlist

# RD Sharma solutions for Class 11 Mathematics chapter 2 - Relations

## Chapter 2: Relations

Ex. 2.10Ex. 2.20Ex. 2.30Others

#### Chapter 2: Relations Exercise 2.10 solutions [Pages 2 - 8]

Ex. 2.10 | Q 1.1 | Page 8

(i) If $\left( \frac{a}{3} + 1, b - \frac{2}{3} \right) = \left( \frac{5}{3}, \frac{1}{3} \right)$ find the values of a and b

Ex. 2.10 | Q 1.2 | Page 8

(ii) If (x + 1, 1) = (3, y − 2), find the values of x and y.

Ex. 2.10 | Q 2 | Page 8

If the ordered pairs (x, −1) and (5, y) belong to the set {(ab) : b = 2a − 3}, find the values of x and y

Ex. 2.10 | Q 3 | Page 8

If a ∈ [−1, 2, 3, 4, 5] and b ∈ [0, 3, 6], write the set of all ordered pairs (ab) such that a + b= 5.

Ex. 2.10 | Q 4 | Page 8

If a ∈ [2, 4, 6, 9] and b ∈ [4, 6, 18, 27], then form the set of all ordered pairs (ab) such that a divides b and a < b.

Ex. 2.10 | Q 5 | Page 8

If A = {1, 2} and B = {1, 3}, find A × B and B × A.

Ex. 2.10 | Q 6 | Page 8

Let A = {1, 2, 3} and B = {3, 4}. Find A × B and show it graphically.

Ex. 2.10 | Q 7 | Page 8

If A = {1, 2, 3} and B = {2, 4}, what are A × BB × AA × AB × B and (A × B) ∩ (B × A)?

Ex. 2.10 | Q 8 | Page 8

If A and B are two set having 3 elements in common. If n(A) = 5, n(B) = 4, find n(A × B) and n[(A × B) ∩ (B × A)].

Ex. 2.10 | Q 9 | Page 8

Let A and B be two sets. Show that the sets A × B and B × A have elements in common iff the sets A and B have an elements in common.

Ex. 2.10 | Q 10 | Page 8

Let A and B be two sets such that n(A) = 3 and n(B) = 2.
If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where xyz are distinct elements.

Ex. 2.10 | Q 11 | Page 8

Let A = {1, 2, 3, 4} and R = {(ab) : a ∈ Ab ∈ Aa divides b}. Write R explicitly.

Ex. 2.10 | Q 12 | Page 8

If A = {−1, 1}, find A × A × A.

Ex. 2.10 | Q 13.1 | Page 8

State whether of  the statement is true or false. If the statement is false, re-write the given statement correctly:

If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}

• True

• False

Ex. 2.10 | Q 13.2 | Page 8

State whether of  the statement is true or false. If the statement is false, re-write the given statement correctly:

If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ B and y ∈ A.

• True

• False

Ex. 2.10 | Q 13.3 | Page 8

State whether of  the statement is true or false. If the statement is false, re-write the given statement correctly:

(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ ϕ) = ϕ.

• True

• False

Ex. 2.10 | Q 14 | Page 8

If A = {1, 2}, from the set A × A × A.

Ex. 2.10 | Q 15.1 | Page 2

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:
(i) A × B

Ex. 2.10 | Q 15.2 | Page 8

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(ii) B × A

Ex. 2.10 | Q 15.3 | Page 8

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(iii) A × A

Ex. 2.10 | Q 15.4 | Page 8

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(iv) B × B

#### Chapter 2: Relations Exercise 2.20 solutions [Page 12]

Ex. 2.20 | Q 1 | Page 12

Given A = {1, 2, 3}, B = {3, 4}, C ={4, 5, 6}, find (A × B) ∩ (B × C ).

Ex. 2.20 | Q 2 | Page 12

If A = {2, 3}, B = {4, 5}, C ={5, 6}, find A × (B ∪ C), A × (B ∩ C), (A × B) ∪ (A × C).

Ex. 2.20 | Q 3.1 | Page 12

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

Ex. 2.20 | Q 3.2 | Page 12

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

Ex. 2.20 | Q 3.3 | Page 12

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:

(iii) A × (B − C) = (A × B) − (A × C)

Ex. 2.20 | Q 4.1 | Page 12

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:

(i) A × C ⊂ B × D

Ex. 2.20 | Q 4.2 | Page 12

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

Ex. 2.20 | Q 5.1 | Page 12

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) A × (B ∩ C)

Ex. 2.20 | Q 5.2 | Page 12

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(ii) (A × B) ∩ (A × C)

Ex. 2.20 | Q 5.3 | Page 12

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(iii) A × (B ∪ C)

Ex. 2.20 | Q 5.4 | Page 12

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(iv) (A × B) ∪ (A × C)

Ex. 2.20 | Q 6 | Page 12

Prove that:

(i)  (A ∪ B) × C = (A × C) ∪ (B × C)

(ii) (A ∩ B) × C = (A × C) ∩ (B×C)

Ex. 2.20 | Q 7 | Page 12

If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.

#### Chapter 2: Relations Exercise 2.30 solutions [Pages 20 - 21]

Ex. 2.30 | Q 1 | Page 20

If A = [1, 2, 3], B = [4, 5, 6], which of the following are relations from A to B? Give reasons in support of your answer.

(i) [(1, 6), (3, 4), (5, 2)]
(ii) [(1, 5), (2, 6), (3, 4), (3, 6)]
(iii) [(4, 2), (4, 3), (5, 1)]
(iv) A × B.

Ex. 2.30 | Q 2 | Page 20

A relation R is defined from a set A = [2, 3, 4, 5] to a set B = [3, 6, 7, 10] as follows:
(xy) ∈ R ⇔ x is relatively prime to y
Express R as a set of ordered pairs and determine its domain and range.

Ex. 2.30 | Q 3 | Page 20

Let A be the set of first five natural numbers and let R be a relation on A defined as follows:
(xy) ∈ R ⇔ x ≤ y
Express R and R−1 as sets of ordered pairs. Determine also (i) the domain of R−1 (ii) the range of R.

Ex. 2.30 | Q 4.1 | Page 20

Find the inverse relation R−1 in each of the cases:

(i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

Ex. 2.30 | Q 4.2 | Page 20

Find the inverse relation R−1 in each of the cases:

(ii) R = {(xy), : xy ∈ N, x + 2y = 8}

Ex. 2.30 | Q 4.3 | Page 20

Find the inverse relation R−1 in each of the cases:

(iii) R is a relation from {11, 12, 13} to (8, 10, 12] defined by y = x − 3.

Ex. 2.30 | Q 5.1 | Page 20

Write the relation as the sets of ordered pairs:

(i) A relation R from the set [2, 3, 4, 5, 6] to the set [1, 2, 3] defined by x = 2y.

Ex. 2.30 | Q 5.2 | Page 20

Write the relation as the sets of ordered pairs:

(ii) A relation R on the set [1, 2, 3, 4, 5, 6, 7] defined by (xy) ∈ R ⇔ x is relatively prime to y.

Ex. 2.30 | Q 5.3 | Page 20

Write the relation as the sets of ordered pairs:

(iii) A relation R on the set [0, 1, 2, ....., 10] defined by 2x + 3y = 12.

Ex. 2.30 | Q 5.4 | Page 20

Write the relation as the sets of ordered pairs:

(iv) A relation R from a set A = [5, 6, 7, 8] to the set B = [10, 12, 15, 16,18] defined by (xy) ∈ R ⇔ x divides y.

Ex. 2.30 | Q 6 | Page 20

Let R be a relation in N defined by (xy) ∈ R ⇔ x + 2y =8. Express R and R−1 as sets of ordered pairs.

Ex. 2.30 | Q 7 | Page 21

Let A = (3, 5) and B = (7, 11). Let R = {(ab) : a ∈ A, b ∈ B, a − b is odd}. Show that R is an empty relation from A into B.

Ex. 2.30 | Q 8 | Page 21

Let A = [1, 2] and B = [3, 4]. Find the total number of relation from A into B.

Ex. 2.30 | Q 9.1 | Page 21

Determine the domain and range of the relation R defined by

(i) R = [(xx + 5): x ∈ (0, 1, 2, 3, 4, 5)]

Ex. 2.30 | Q 9.2 | Page 21

Determine the domain and range of the relation R defined by

(ii) R = {(xx3) : x is a prime number less than 10}

Ex. 2.30 | Q 10.1 | Page 21

Determine the domain and range of the relations:

(i) R = {(ab) : a ∈ N, a < 5, b = 4}

Ex. 2.30 | Q 10.2 | Page 21

Determine the domain and range of the relations:

(ii) $S = \left\{ \left( a, b \right) : b = \left| a - 1 \right|, a \in Z \text{ and} \left| a \right| \leq 3 \right\}$

Ex. 2.30 | Q 11 | Page 21

Let A = {ab}. List all relations on A and find their number.

Ex. 2.30 | Q 12 | Page 21

Let A = (xyz) and B = (ab). Find the total number of relations from A into B.

Ex. 2.30 | Q 13.1 | Page 21

Let R be a relation from N to N defined by R = [(ab) : ab ∈ N and a = b2]. Are the statement true?

(i) (aa) ∈ R for all a ∈ N

Ex. 2.30 | Q 13.2 | Page 21

Let R be a relation from N to N defined by R = [(ab) : ab ∈ N and a = b2]. Are the statement true?

(ii) (ab) ∈ R ⇒ (ba) ∈ R

Ex. 2.30 | Q 13.3 | Page 21

Let R be a relation from N to N defined by R = [(ab) : ab ∈ N and a = b2]. Are the statement true?

(iii) (ab) ∈ R and (bc) ∈ R ⇒ (ac) ∈ R

Ex. 2.30 | Q 14 | Page 21

Let A = [1, 2, 3, ......., 14]. Define a relation on a set A by
R = {(xy) : 3x − y = 0, where xy ∈ A}.
Depict this relationship using an arrow diagram. Write down its domain, co-domain and range.

Ex. 2.30 | Q 15 | Page 21

Define a relation R on the set N of natural number by R = {(xy) : y = x + 5, x is a natural number less than 4, xy ∈ N}. Depict this relationship using (i) roster form (ii) an arrow diagram. Write down the domain and range or R.

Ex. 2.30 | Q 16 | Page 21

A = [1, 2, 3, 5] and B = [4, 6, 9]. Define a relation R from A to B by R = {(xy) : the difference between x and y is odd, x ∈ A, y ∈ B}. Write R in Roster form.

Ex. 2.30 | Q 17 | Page 21

Write the relation R = {(xx3) : x is a prime number less than 10} in roster form.

Ex. 2.30 | Q 18 | Page 21

Let A = [1, 2, 3, 4, 5, 6]. Let R be a relation on A defined by {(ab) : ab ∈ A, b is exactly divisible by a}

(i) Writer R in roster form
(ii) Find the domain of R
(ii) Find the range of R.

Ex. 2.30 | Q 19 | Page 21

The adjacent figure shows a relationship between the sets P and Q. Write this relation in (i) set builder form (ii) roster form. What is its domain and range?

Ex. 2.30 | Q 20 | Page 21

Let R be the relation on Z defined by
R = {(ab) : ab ∈ Z, a − b is an integer}
Find the domain and range of R.

Ex. 2.30 | Q 21 | Page 21

For the relation R1 defined on R by the rule (ab) ∈ R1 ⇔ 1 + ab > 0. Prove that: (ab) ∈ R1 and (b , c) ∈ R1 ⇒ (ac) ∈ R1 is not true for all abc ∈ R.

Ex. 2.30 | Q 22.1 | Page 21

Let R be a relation on N × N defined by
(ab) R (cd) ⇔ a + d = b + c for all (ab), (cd) ∈ N × N
Show that:
(i) (ab) R (ab) for all (ab) ∈ N × N

Ex. 2.30 | Q 22.2 | Page 21

Let R be a relation on N × N defined by
(ab) R (cd) ⇔ a + d = b + c for all (ab), (cd) ∈ N × N
Show that:

(ii) (ab) R (cd) ⇒ (cd) R (ab) for all (ab), (cd) ∈ N × N

Ex. 2.30 | Q 22.3 | Page 21

Let R be a relation on N × N defined by
(ab) R (cd) ⇔ a + d = b + c for all (ab), (cd) ∈ N × N

(iii) (ab) R (cd) and (cd) R (ef) ⇒ (ab) R (ef) for all (ab), (cd), (ef) ∈ N × N

#### Chapter 2: Relations solutions [Pages 24 - 25]

Q 1 | Page 24

If A = {1, 2, 4}, B = {2, 4, 5} and C = {2, 5}, write (A − C) × (B − C).

Q 2 | Page 24

If n(A) = 3, n(B) = 4, then write n(A × A × B).

Q 3 | Page 24

If R is a relation defined on the set Z of integers by the rule (xy) ∈ R ⇔ x2 + y2 = 9, then write domain of R.

Q 4 | Page 25

If R = {(xy) : xy ∈ Z, x2 + y2 ≤ 4} is a relation defined on the set Z of integers, then write domain of R.

Q 5 | Page 25

If R is a relation from set A = (11, 12, 13) to set B = (8, 10, 12) defined by y = x − 3, then write R−1.

Q 6 | Page 25

Let A = {1, 2, 3} and$R = \left\{ \left( a, b \right) : \left| a^2 - b^2 \right| \leq 5, a, b \in A \right\}$.Then write R as set of ordered pairs.

Q 7 | Page 25

Let R = [(xy) : xy ∈ Z, y = 2x − 4]. If (a, -2) and (4, b2) ∈ R, then write the values of a and b.

Q 8 | Page 25

If R = {(2, 1), (4, 7), (1, −2), ...}, then write the linear relation between the components of the ordered pairs of the relation R.

Q 9 | Page 25

If A = [1, 3, 5] and B = [2, 4], list of elements of R, if
R = {(xy) : xy ∈ A × B and x > y}

Q 10 | Page 25

If R = [(xy) : xy ∈ W, 2x + y = 8], then write the domain and range of R.

Q 11 | Page 25

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, write A and B

Q 12 | Page 25

Let A = [1, 2, 3, 5], B = [4, 6, 9] and R be a relation from A to B defined by R = {(xy) : x − yis odd}. Write R in roster form.

#### Chapter 2: Relations solutions [Pages 25 - 26]

Q 1 | Page 25

If A = {1, 2, 4}, B = {2, 4, 5}, C = {2, 5}, then (A − B) × (B − C) is

• (a) {(1, 2), (1, 5), (2, 5)}

• (b) [(1, 4)]

• (c) (1, 4)

• (d) none of these

Q 2 | Page 25

If R is a relation on the set A = [1, 2, 3, 4, 5, 6, 7, 8, 9] given by x R y ⇔ y = 3x, then R =

• (a) [(3, 1), (6, 2), (8, 2), (9, 3)]

• (b) [(3, 1), (6, 2), (9, 3)]

• (c) [(3, 1), (2, 6), (3, 9)]

• (d) none of these

Q 3 | Page 25

Let A = [1, 2, 3], B = [1, 3, 5]. If relation R from A to B is given by = {(1, 3), (2, 5), (3, 3)}, Then R−1 is

• (a) {(3, 3), (3, 1), (5, 2)}

• (b) {(1, 3), (2, 5), (3, 3)}

• (c) {(1, 3), (5, 2)}

• (d) None of these

Q 4 | Page 25

If A = [1, 2, 3], B = [1, 4, 6, 9] and R is a relation from A to B defined by 'x' is greater than y. The range of R is

• (a) {1, 4, 6, 9}

• (b) (4, 6, 9)

• (c) [1]

• (d) none of these.

Q 5 | Page 25

If R = {(xy) : xy ∈ Z, x2 + y2 ≤ 4} is a relation on Z, then the domain of R is

• (a) [0, 1, 2]

• (b) [0, −1, −2]

• (c) {−2, −1, 0, 1, 2]

• (d) None of these

Q 6 | Page 25

A relation R is defined from [2, 3, 4, 5] to [3, 6, 7, 10] by : x R y ⇔ x is relatively prime to y. Then, domain of R is

• (a) [2, 3, 5]

• (b) [3, 5]

• (c) [2, 3, 4]

• (d) [2, 3, 4, 5]

Q 7 | Page 26

A relation ϕ from C to R is defined by x ϕ y ⇔ |x| = y. Which one is correct?

• (a) (2 + 3i) ϕ 13

• (b) 3ϕ (−3)

• (c) (1 + i) ϕ 2

• (d) i ϕ 1

Q 8 | Page 26

Let R be a relation on N defined by x + 2y = 8. The domain of R is

• (a) [2, 4, 8]

• (b) [2, 4, 6, 8]

• (c) [2, 4, 6]

• (d) [1, 2, 3, 4]

Q 9 | Page 26

R is a relation from [11, 12, 13] to [8, 10, 12] defined by y = x − 3. Then, R−1 is

• (a) [(8, 11), (10, 13)]

• (b) [(11, 8), (13, 10)]

• (c) [(10, 13), (8, 11), (12, 10)]

• (d) none of these

Q 10 | Page 26

If the set A has p elements, B has q elements, then the number of elements in A × B is

• (a) p + q

• (b) p + q + 1

• (c) pq

• (d) p2

Q 11 | Page 26

Let R be a relation from a set A to a set B, then

• (a) R = A ∪ B

• (b) R = A ∩ B

• (c) R ⊆ A × B

• (d) R ⊆ B × A

Q 12 | Page 26

If R is a relation from a finite set A having m elements of a finite set B having n elements, then the number of relations from A to B is

• (a) 2mn

• (b) 2mn − 1

• (c) 2mn

• (d) mn

Q 13 | Page 26

If R is a relation on a finite set having n elements, then the number of relations on A is

• (a) 2n

• (b)  $2^{n^2}$

• (c) n2

• (d) nn

## Chapter 2: Relations

Ex. 2.10Ex. 2.20Ex. 2.30Others

## RD Sharma solutions for Class 11 Mathematics chapter 2 - Relations

RD Sharma solutions for Class 11 Maths chapter 2 (Relations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics chapter 2 Relations are Brief Review of Cartesian System of Rectanglar Co-ordinates, Logarithmic Functions, Exponential Function, Pictorial Representation of a Function, Graph of Function, Pictorial Diagrams, Equality of Ordered Pairs, Ordered Pairs, Algebra of Real Functions, Some Functions and Their Graphs, Functions, Relation, Cartesian Product of Sets.

Using RD Sharma Class 11 solutions Relations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 2 Relations Class 11 extra questions for Maths and can use Shaalaa.com to keep it handy for your exam preparation

S