11th NCERT Relatins and functions.Exercise 2.1 Questions 10
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Question (1)

If,$\left( {\frac{x}{3} + 1,y - \frac{2}{3}} \right) = \left( {\frac{5}{3},\frac{1}{3}} \right)$ find the values of x and y.

Solution

$\left( {\frac{x}{3} + 1,y - \frac{2}{3}} \right) = \left( {\frac{5}{3},\frac{1}{3}} \right)$ $\Rightarrow \frac{x}{3} + 1 = \frac{5}{3},\quad y - \frac{2}{3} = \frac{1}{3}$ $\Rightarrow \frac{x}{3} = \frac{5}{3} - 1,\quad y\frac{2}{3} = \frac{1}{3} + \frac{2}{3}$ $x = \frac{2}{3},y = 1$

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)?

Solution

n(A) = 3, n(B) = 3, find n(A × B)
n(A × B) = n(A) × n(B)
= 3 × 3
=9

Question (3)

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

Solution

G = {7, 8} and H = {5, 4, 2}
We know that the Cartesian product P × Q of two non-empty sets P and Q is defined as
P × Q = {(p, q): p ∈ P, q ∈ Q}
∴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 statement are true or false. If the statement is false, rewrite the given statement correctly.
(i) If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.

Solution

The statement is false.
If P = {m, n} and Q = {n, m}, then P × Q = {(m, n),(m, m),(n, 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 ∈ A and y ∈ B.

Solution

The statement is true.
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ.

Solution

The statement is true.

Question (5)

If A = {–1, 1}, find A × A × A.

Solution

It is known that for any non-empty set A, A × A × A is defined as
A × A × A = {(a, b, c): a, b, c ∈ A}
It is given that A = {–1, 1}
∴ 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.

Solution

We know that A × B = {( x, y) :x ∈ A and y ∈ B}
A × B = {(a, x), (a, y), (b, x), (b, y)}
So A = { a, b} and B = { x, y}

Question (7)

Let A = {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

Solution

(i) A × (B ∩ C) = (A × B) ∩ (A × C)
B = {1, 2, 3, 4}, C = {5, 6}
B ∩ C = {} = φ
LHS = A × (B ∩ C)
= A × φ
= φ
A = {1, 2}, B = {1, 2, 3, 4},
So A × B = { (1, 1), (1, 2),(1, 3), (1, 4), (2, 1),(2, 2), (2, 3), (2, 4)}
A = {1, 2}, C = {5, 6}
So A × C = { (1, 5), (1, 6), (2, 5), (2, 6)}
RHS =( A × B) ∩ (A × C)
= φ
= LHS
So A × (B ∩ C) = (A × B) ∩ (A × C)
(ii) A × C is a subset of B × D
A = {1, 2}, C = {5, 6}
So A × C = { (1, 5), (1, 6), (2, 5), (2, 6)}
B = {1, 2, 3, 4}, D = {5, 6, 7, 8}.
So 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)}
Each and every element of A × C is present in B × D
So 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.

Solution

A = {1, 2} and B = {3, 4}.
A × B = {(1,3),(1,4),(2,3),(2,4)}
Number of sub set of A × B = 24 = 16
The sub sets of A × B are
φ, {(1,3)}, {(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)} {(1,4),(2,3),(2,4)}, {(1,3),(1,4),(2,3),(2,4)}

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.

Solution

n(A) = 3 and n (B) = 2.
If (x, 1), (y, 2), (z, 1) are in A × B,
A = { x, y, z} and 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.

Solution

n(A × A = 9
n(A) × n(A) = 9
∴ n(A) = 3
(–1, 0) and (0, 1). ∈ A × A
So A = { -1, 0, 1}
∴ A × A = {(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}
So remaining elements of A × A are
{(-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)}