Relations and Functions

CBSE Class–11 Mathematics

NCERT Solutions
Chapter - 2 Relations and Functions
Exercise 2.1

1. If find the values of and

Ans. Here

and

2. If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A B).

Ans. Number of elements in set A = 3 and Number of elements in set B = 3

Number of elements in A B = 3 3 = 9

3. If G = {7, 8} and H = {5, 4, 2}, find G H and H G.

Ans. Given: G = {7, 8} and H = {5, 4, 2}

GH = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

And H G = {(5, 7), (4, 7), (2, 7), (5, 8), (4, 8), (2, 8)}

4. State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly:

(i) If P = and Q = then P Q =

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

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

Ans. (i) Here P = and Q =

Number of elements in set P = 2 and Number of elements in set Q = 2

Number of elements in P Q = 2 2 = 4

But PQ = and here number of elements in P Q = 2

Therefore, statement is false.

Correct statment is $P \times Q = {(m, m), (n, n), (n, m), (m, n)}$

(ii) True

(iii) True

5. If A = find A A A.

Ans. Here A =

A A =

A A A =

6. If A B = find A and B.

Ans. Given: A B =

A = set of first elements = and B = set of second elements =

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

(i)

(ii) A C is a subset of B D.

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

C = {5, 6} and D = {5, 6, 7, 8}

(i) = {1, 2, 3, 4} {5, 6} =

……….(i)

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

A C = {(1, 5), (1, 6), (2, 5), (2, 6)

(AB) (A C) = ……….(ii)

Therefore, from eq. (i) and (ii),

= (A B) (A C)

(ii) A C = {(1, 5), (1, 6), (2, 5), (2, 6)

B D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8),

(4, 5), (4, 6), (4, 7), (4, 8),

Therefore, it is clear that each element of A C is present in B D.

A C B D

8. Let A = {1, 2} and B = {3, 4}, write A B. How many subsets will A B have? List them.

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

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

Number of elements in A B = 4

Therefore, Number of subsets of AB=24 =16

$ϕ$ ,{(2,3)},{(1,4)},{(2,3)},{(2,4)},{(1, 3), (1, 4)} ,{(1, 3),(2, 3)} ,{(1, 3),(2, 4)},{(1, 4), (2, 3)} ,{(1, 4),(2, 4)},{(2, 3), (2, 4)},{(1, 3), (1, 4), (2, 3)} ,{(1, 3), (1, 4), (2, 4)} ,{(1, 3), (2, 3), (2, 4)} ,{(1, 4), (2, 3), (2, 4)},{(1, 3), (1, 4), (2, 3), (2, 4)}

9. Let A and B be two sets such that and If are in A B.