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# RD Sharma solutions for Class 11 Mathematics Textbook chapter 2 - Relations [Latest edition]

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## Chapter 2: Relations

Ex. 2.1Ex. 2.2Ex. 2.3Others

#### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.1 [Pages 2 - 8]

Ex. 2.1 | 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. 2.1 | Q 1.2 | Page 8

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

Ex. 2.1 | 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. 2.1 | 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. 2.1 | 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. 2.1 | Q 5 | Page 8

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

Ex. 2.1 | Q 6 | Page 8

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

Ex. 2.1 | 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. 2.1 | 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. 2.1 | 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. 2.1 | 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. 2.1 | Q 11 | Page 8

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

Ex. 2.1 | Q 12 | Page 8

If A = {−1, 1}, find A × A × A.

Ex. 2.1 | 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. 2.1 | 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. 2.1 | 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. 2.1 | Q 14 | Page 8

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

Ex. 2.1 | Q 15.1 | Page 2

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

Ex. 2.1 | Q 15.2 | Page 8

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

(ii) B × A

Ex. 2.1 | Q 15.3 | Page 8

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

(iii) A × A

Ex. 2.1 | Q 15.4 | Page 8

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

(iv) B × B

#### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.2 [Page 12]

Ex. 2.2 | Q 1 | Page 12

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

Ex. 2.2 | 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. 2.2 | 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. 2.2 | 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. 2.2 | 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. 2.2 | 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. 2.2 | 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. 2.2 | Q 5.1 | Page 12

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

(i) A × (B ∩ C)

Ex. 2.2 | 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. 2.2 | Q 5.3 | Page 12

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

(iii) A × (B ∪ C)

Ex. 2.2 | 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. 2.2 | Q 6 | Page 12

Prove that:

(i)  (A ∪ B) × C = (A × C) ∪ (B × C)

(ii) (A ∩ B) × C = (A × C) ∩ (B×C)

Ex. 2.2 | Q 7 | Page 12

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

#### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.3 [Pages 20 - 21]

Ex. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | Q 8 | Page 21

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

Ex. 2.3 | 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. 2.3 | 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. 2.3 | Q 10.1 | Page 21

Determine the domain and range of the relations:

(i) R = {(ab) : a ∈ N, a < 5, b = 4}

Ex. 2.3 | 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. 2.3 | Q 11 | Page 21

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

Ex. 2.3 | Q 12 | Page 21

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

Ex. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | Q 17 | Page 21

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

Ex. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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. 2.3 | 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

#### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations [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.

#### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations [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. 2.1Ex. 2.2Ex. 2.3Others
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## RD Sharma solutions for Class 11 Mathematics Textbook chapter 2 - Relations

RD Sharma solutions for Class 11 Mathematics Textbook 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 Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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

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