# Relations and Functions

## NCERT Solution

### Exercise 2.1

Question 1: If `(x/3+1, y-2/3)=(5/3, 1/3)` find the values of x and y.

**Answer:** `x/3+1=5/3`

Or, `x/3=5/3-1=(5-3)/3`

Or, `x/3=2/3`

Or, `x=2`

Similarly, `y-2/3=1/3`

Or, `y=1/3+2/3`

Or, `y=(1+2)/3=3/3`

Or, `y=1`

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

**Answer:** n(A) = 3 and n(B) = 3

So, n(A × B) = 3 × 3 = 9

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

**Answer:** G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

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

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

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

**Answer:** This is a false statement. Correct statement is as follows:

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

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

**Answer:**True

(c) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ

**Answer:** True

Question 5: If A = {-1, 1}, find A × A × A

**Answer:** 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)}

Question 6: If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.

**Answer:** A = {a, b} and B = {x, y}

Question 7: Let ={1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that (i) A × (B ∪ C) = (A × B) ∪ (A × C). (ii) A × C is a subset of B × D.

**Answer (i):** A × (B ∩ C) = A × Φ = Φ……….. (1)

(A × B) ∩ (A × C)

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

= Phi; ………. (2)

From equations (1) and (2), it is clear, A × (B ∪ C) = (A × B) ∪ (A × C)

**Answer (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)}

From above equations, it is clear that A × C is a subset of B × D

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

**Answer:** A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}

There are 4 subsets

Question 9: 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 and z are distinct elements.

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

So, `n(A) × n(B) = 3 × 2 = 6`

Given members of A × B indicate the following:

A = {x, y, z}

B = {1, 2}

Question 10: The Cartesian product A × A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A × A.

**Answer:** A = {-1, 0, 1}