Students often refer to NCERT Class 9 Advanced Maths Solutions Chapter 3 Relations and Functions Ex 3.1 to verify their answers.
Advanced Maths Class 9 Exercise 3.1 Solutions
Class 9 Advanced Maths Ex 3.1 Solutions
Question 1.
If (x – 5, y + 1) = (4, 6), find x and y.
Solution:
Since ordered pairs are equal, corresponding elements are equal.
x – 5 = 4
x = 9
and y + 1 = 6
y = 5
x = 9,y = 5
Therefore, the values of x and y are 9 and 5, respectively.
Question 2.
Let A = (1, 2} and B = {2, 3, 5}. List all elements of A × B and B × A.
Solution:
A × B = {(a, b): a ∈ A, b ∈ B}
∴ A × B = {(1, 2), (1, 3), (1, 5), (2, 2), (2, 3), (2, 5)}
B × A = {(b, a): b ∈ B, a ∈ A}
∴ B × A = {(2, 1), (2, 2), (3, 1), (3, 2), (5, 1), (5, 2)}
Question 3.
If n(A × B) = 20 and n(A) = 4, find n(B).
Solution:
We know,
n(A × B) = n(A) × n(B)
20 = 4 × n(B)
n(B) = \(\frac{20}{4}\) = 5
n(B) = 5
Question 4.
If A = {1, 2, 3} and B = {x, y}, find A × B, B × A, A × A and B × B.
Solution:
(i) A × B = {(a, b) :a ∈ A, b ∈ B}
∴ A × B = {(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)}
(ii) B × A = {(b, a): b ∈ B, a ∈ A}
B × A = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}
(iii) A × A = {(a, a): a ∈ A, a ∈ A}
∴ A × A = {(1,1), (1, 2), (1, 3), (2,1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
(iv) B × B = {(b, b): b ∈ B, b ∈ B}
∴ B × B = {(x, x), (x, y), (y, x), (y, y)}
Question 5.
If A = {1, 2, 3} and B = {2, 3, 7}, find (A × B) ∩ (B × A).
Solution:
First, we find common ordered pairs.
A × B = {(a, b) :a ∈ A, b ∈ B}
∴ A × B = {(1, 2), (1, 3), (1, 7), (2,2), (2, 3), (2, 7), (3, 2), (3, 3), (3,7)}
B × A = {(b, a) :b ∈ B, a ∈ A}
∴ B × A = {(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (7, 1), (7, 2), (7, 3)}
Common ordered pairs are: (2, 2), (2, 3), (3, 2), (3, 3)
Hence, (A × B) ∩ (B × A) = {(2, 2), (2, 3), (3, 2), (3, 3)}
Question 6.
Verify, A × (B ∪ C) = (A × B) ∪ (A × C) for A = {1, 2}, B = {2, 3}, C = {4, 5}.
Solution:
First, B ∪ C = {2, 3, 4, 5}
LHS:
A × (B ∪ C)
= {(1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5)}
RHS:
A × B = {(1, 2), (1, 3), (2, 2), (2, 3)}
A × C = {(1, 4), (1, 5), (2, 4), (2, 5)}
Their union:
(A × B) ∪ (A × C)
= {(1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5)}
Since both sides are equal,
A × (B ∪ C) = (A × B) ∪ (A × C) Hence Verified.