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

(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. 

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

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

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

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

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

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

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

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)].

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. 

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 x, y, z are distinct elements.

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

If A = {−1, 1}, find A × A × A.

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)}

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.

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

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

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

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

(ii) B × A

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

(iii) A × A

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

(iv) B × B 

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

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

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

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

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

(iii) A × (B − C) = (A × B) − (A × C)

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

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

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

(i) A × (B ∩ C)

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

(ii) (A × B) ∩ (A × C)

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

(iii) A × (B ∪ C)

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

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.

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

Let A be the set of first five natural numbers and let R be a relation on A defined as follows:
(x, y) ∈ 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.

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

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

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

(ii) R = {(x, y), : x, y ∈ N, x + 2y = 8}

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

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.

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 (x, y) ∈ R ⇔ x is relatively prime to y.

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.

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 (x, y) ∈ R ⇔ x divides y.

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

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

Determine the domain and range of the relation R defined by

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

Determine the domain and range of the relation R defined by

Determine the domain and range of the relations:

(i) R = {(a, b) : a ∈ N, a < 5, b = 4}

Determine the domain and range of the relations:

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

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Is the following true?

(a, a) ∈ R, for all a ∈ N

Justify your answer in case.

Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b²}. Is the statement true?

(a, b) ∈ R implies (b, a) ∈ R

Justify your answer in case.

Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b²}. Is the statement true?

(a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R

Justify your answer in case.

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

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

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

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

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?

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

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

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

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

(ii) (a, b) R (c, d) ⇒ (c, d) R (a, b) for all (a, b), (c, d) ∈ N × N

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

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

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

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

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.

 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.

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

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

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

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

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

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

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

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 =

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

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

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

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

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

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

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

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

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

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