NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Ex 2.1
NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Ex 2.1
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Solutions for Class 11 Maths Chapter 2 Relations and Functions Ex 2.1.
Ex 2.1 Class 11 Maths Question 1.
If (x/3 + 1, y − 2/3) = (5/3, 1/3), find the values of x and y.Solution.
If the ordered pairs are equal, the corresponding elements are also equal.
∴ x/3 + 1 = 5/3 and y − 2/3 = 1/3
⇒ x/3 = 5/3 − 1 and y = 1/3 + 2/3
⇒ x = 2 and
y = 1.
Ex 2.1 Class 11 Maths 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.
According to the question, n(A) = 3 and n(B) = 3.
∴ n(A × B) = n(A) × n(B) = 3 × 3 = 9
∴ There are a total of 9 elements in (A ×
B).
Ex 2.1 Class 11 Maths Question 3.
If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.Solution.
We have G = {7, 8} and H = {5, 4, 2}. Then, by the definition of the cartesian
product, we have
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)}.
Ex 2.1 Class 11 Maths Question 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 = {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 ∈ A and y ∈ B.
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ
Solution.
(i) False,
if P = {m, n} and Q = {n, m}
Then P × Q = {(m, n), (m, m), (n, n), (n, m)}.
(ii) True,
by the definition of cartesian product.
(iii) True,
we have A = {1, 2} and B = {3, 4}
Now, B ∩ φ = φ
∴ A × (B ∩ φ) = A × Ï† = φ
Ex 2.1 Class 11 Maths Question 5.
If A = {-1, 1}, find A × A × A.Solution.
A = {-1, 1}
Then, A × A = {-1, 1} × {-1, 1} = {(-1, -1), (-1, 1), (1, -1), (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, -1), (1, -1, 1),
(1, 1, -1), (1, 1, 1)}
Ex 2.1 Class 11 Maths Question 6.
If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.Solution.
Given, A × B = {(a, x), (a, y), (b, x), (b, y)}
If (p, q) ∈ A × B, then p ∈ A and q ∈ B
∴ A = {a, b} and B = {x, y}.
Ex 2.1 Class 11 Maths 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.
Given, A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, D = {5, 6, 7, 8}
Ex 2.1 Class 11 Maths Question 8.
Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.Solution.
Given, A = {1, 2} and B = {3, 4}
Then, A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}
i.e., A × B has 4 elements. So, it has 24, i.e., 16 subsets.
The subsets of A × B are as follows:
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), (2,
3), (2, 4)}, {(1, 3), (1, 4), (2, 3), (2, 4)}
Ex 2.1 Class 11 Maths 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.
Given, n(A) = 3 and n(B) = 2.
Now (x, 1) ∈ A × B ⇒ x ∈
A and 1 ∈ B,
(y, 2) ∈ A × B ⇒ y ∈
A and 2 ∈ B
(z, 1) ∈ A × B ⇒ z ∈
A and 1 ∈ B
∴ x, y, z ∈ A and 1, 2 ∈ B
Hence, A = {x, y, z} and B = {1, 2}.
Ex 2.1 Class 11 Maths 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.
Since, we have n(A × A) = 9
⇒ n(A) × n(A) = 9 [ ∵ n(A × B) = n(A) × n(B)]
⇒ (n(A))2 = 9 ⇒ n(A) = 3
Also, given (-1, 0) ∈ A × A ⇒ -1, 0 ∈ A ,
and (0, 1) ∈ A × A ⇒ 0, 1 ∈ A
∴ -1, 0, 1 ∈ A
Hence, A = {-1, 0, 1} [∵ n(A) = 3]
The remaining elements of A × A are (-1, -1), (-1, 1), (0, -1), (0, 0), (1,
-1), (1, 0), (1, 1).
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