Class 11 RD Sharma Solutions - Chapter 2 Relations - Exercise 2.1

Chapter 2 of RD Sharma's Class 11 Mathematics textbook focuses on Relations. Exercise 2.1 specifically deals with the basic concepts of relations, including their definition, representation, and properties. This exercise helps students understand how relations are used to describe connections between elements of sets.

Key concepts covered in this exercise include:

• Definition of a relation
• Representing relations using set notation, arrow diagrams, and graphs
• Domain and range of a relation
• Types of relations (empty, universal, identity)

Question 1(i). If (a/3 + 1, b - 2/3) = (5/3, 1/3), find the values of a and b.

Solution:

According to the definition of equality of ordered pairs

 (a/3 + 1, b - 2/3) = (5/3, 1/3)

⇒ a/3 + 1 = 5/3 and b - 2/3 =1/3

⇒ a/3 = (5 - 3)/3 and b = (1/3 + 2/3)

⇒ a/3 = 2/3 and b = 3/3

⇒ a = 2 and b = 1

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

Solution:

According to the definition of equality of ordered pairs

(x + 1, 1) = (3, y - 2)

⇒ x + 1 = 3 and 1 = y - 2

⇒ x = 3 - 1 and 1 + 2 = y

⇒ x = 2 and 3 = y

⇒ x = 2 and y = 3

Question 2. 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.

Solution:

Given:

(x, -1) ∈ {(a, b) : b = 2a - 3}

and, (5, y) ∈ {(a, b) : b = 2a - 3}

⇒ -1 = 2 × x - 3 and y = 2 × 5 - 3 

⇒ -1 = 2x - 3 and y = 10 - 3

⇒ 3 - 1 = 2x and y = 7

⇒ 2 = 2x and y = 7 

⇒ x = 1 and y = 7

Question 3. 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.

Solution:

Given: a ∈ {- 1, 2, 3, 4, 5} and b ∈ {0, 3, 6},

Now, we have to find the ordered pair (a, b) such that a + b = 5

So, the ordered pair (a, b) such that a + b = 5 are as follows

(a, b) ∈ {(- 1, 6), (2, 3), (5, 0)}

Question 4. 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.

Solution:

Given: a ∈ {2, 4, 6, 9} and b ∈ {4, 6, 18, 27}

Here,

2 divides 4, 6, 18 and is also less than all of them

4 divides 4 and is also less than none of them

6 divides 6, 18 and is less than 18 only

9 divides 18, 27 and is less than all of them

∴ Ordered pairs (a, b) are (2, 4), (2, 6), (2, 18), (6, 18), (9, 18) and (9, 27)

Question 5. If A = {1, 2} and B = {1, 3}, find A x B and B x A.

Solution:

Given: A = {1, 2} and B = {1, 3}

Now we have to find A x B, and B x A

A × B = {1, 2} × {1, 3}

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

B × A = {1, 3} × {1, 2}

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

Question 6. Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically

Solution:

Given: A = {1, 2, 3} and B = {3, 4}

Now we have to find A x B

A x B = {1, 2, 3} × {3, 4}

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

To draw A x B graphically follow the following steps:

Step 1: Draw horizontal and vertical axis.

Step 2: The horizontal axis represents set A and the vertical axis represents set B.

Step 3: Now, draw dotted lines perpendicular to horizontal and vertical axes through the elements of set A and B

Step 4: Point of intersection of these perpendicular represents A × B

👁 Image

Question 7. If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B) ∩ (B x A)?

Solution:

Given:

A = {1, 2, 3} and B = {2, 4}

Now we have to find A × B, B × A, A × A, and (A × B) ∩ (B × A)

A × B = {1, 2, 3} × {2, 4}

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

B × A = {2, 4} × {1, 2, 3}

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

A × A = {1, 2, 3} × {1, 2, 3}

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

B × B = {2, 4} × {2, 4}

= {(2, 2), (2, 4), (4, 2), (4, 4)}

Intersection of two sets represents common elements of both the sets

So,

(A × B) ∩ (B × A) = {(2, 2)} 

Question 8. If A and B two sets having 3 elements in common. If n(A) = 5, n(B) = 4, Find n(A × B) and n[(A × B) ∩ (B × A)]

Solution:

Given: n(A) = 5 and n(B) = 4

We know that if A and B are two finite sets, then n(A × B) = n(A) × n(B)

Therefore,

n(A × B) = 5 × 4 = 20

Now,

n[(A × B) ∩ (B × A)] = 3 × 3 = 9          -(∵ A and B have 3 common elements)

Question 9. Let A and B two sets. Show that the sets A × B and B × A has an elements in common if the sets A and B have an element in common.

Solution:

Let us considered(a, b) be an arbitrary elements of (A × B) ∩ (B × A). Then,

(a, b) ∈ (A × B) ∩ (B × A)

= (a, b) ∈ A × B and (a, b) ∈ B × A

= (a ∈ A and b ∈ B) and (a ∈ B and b ∈ A)

= (a ∈ A and a ∈ B) and (b ∈ A and b ∈ B)

= a ∈ A ∩ B and b ∈ A ∩ B 

Hence, the sets A × B and B × A have an element in 

common have an element in common.

Question 10. Let A and B 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

Solution:

Since (x, 1), (y, 2), (z, 1) are elements of A × B. Therefore, x, y, z ∈ A and 1, 2 ∈ B

Given: n(A) = 3 and n(B) = 2

Therefore, x, y, z ∈ A and n(A) = 3

⇒ A = (x, y, z)

1, 2 ∈ B and n(B) = 2

⇒ B = (1, 2)

Question 11. Let A = {1, 2, 3, 4} and R = {(a, b) : a ∈ A, b ∈ A, a divides b}. Write R explicitly.

Solution: 

Given: A = (1, 2, 3, 4) and, R = {(a, b) : a ∈ A, b ∈ A, a divides b}

Now, a/b stands for 'a divides b'. 

So, for the elements of the given sets, we find that 1/1, 1/2, 1/3, 1/4, 2/2, 3/3 and 4/4

Therefore,

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

Question 12. If A = {-1, 1}, find A × A × A

Solution:

Given: A = {-1, 1}

So, A × A = {-1, 1} × {-1, 1}

= {(-1, -1), (-1, 1), (1, -1), (1, 1)}

Therefore, A × A × A = {-1, 1} × {(-1, -1), (-1, 1), (1, -1)(1, 1)}

= {(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)}

Question 13. State whether each of the following statements are true or false. If the statements is false, re-write the given statement correctly:

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

(ii) 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.

(iii) If A = {1, 2} and B = {3, 4}, then A ∩ (B ∩ ∅) = ∅ 

Solution:

(i) False,

If P = {m, n} and Q = {n, m},

Then,

P × Q = {(m, n), (m, m), (n, n), (n, m)}

(ii) False,

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

(iii) True

Question 14. If A = {1, 2} form the set A × A × A

Solution:

Given: A = {1, 2}

So, A × A = {1, 2} × {1, 2}

= {(1, 1), (1, 2), (2, 1), (2, 2)}

Therefore,

A × A × A = {1, 2} × {(1, 1), (1, 2), (2, 1), (2, 2)}

= {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}

Question 15 (i). If A = {1, 2, 4} and B = {1, 2, 3}, represent A × B graphically

Solution:

Given: A = {1, 2, 4} and B = {1, 2, 3}

So, A × B = {1, 2, 4} × {1, 2, 3}

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

Hence, we represent A on the horizontal line and B on vertical line.

So, the graphical representation of A × B is as shown below:

👁 Image

Question 15 (ii). If A = {1, 2, 4} and B = {1, 2, 3}, represent B × A graphically.

Solution:

Given: A = {1, 2, 4} and B = {1, 2, 3}

So, B × A = {1, 2, 3} × {1, 2, 4}

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

Hence, we represent B on the horizontal line and A on vertical line.

So, the graphical representation of B × A:

👁 Image

Question 15 (iii). If A = {1, 2, 4} and B = {1, 2, 3}, represent A × A graphically.

Solution:

Given: A = {1, 2, 4}, B = {1, 2, 3}

So, A × A = {1, 2, 4} × {1, 2, 4}

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

Graphical representation of A × A:

👁 Image

Question 15 (iv). If A = {1, 2, 4} and B = {1, 2, 3}, represent B × B graphically.

Solution:

Given: A = {1, 2, 4}, B = {1, 2, 3}

So, B × B = {1, 2, 3} × {1, 2, 3}

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

Graphical representation of B × B:

Summary

Chapter 2 Exercise 2.1 in RD Sharma's Class 11 Mathematics textbook focuses on the fundamental concepts of relations. It introduces students to the definition of relations and various methods of representing them, including set notation, arrow diagrams, and graphs. The exercise covers key topics such as identifying the domain and range of a relation, understanding different types of relations (empty, universal, identity), and analyzing properties like reflexivity, symmetry, and transitivity. Students learn to work with operations on relations, including union, intersection, and composition. This foundational knowledge is crucial for understanding more advanced topics in relations and functions, which play a significant role in higher mathematics and its real-world applications. The exercise provides a mix of theoretical understanding and practical problem-solving, preparing students for more complex mathematical concepts.