NCERT Solutions Class 11 Mathematics Relation Functions Exercise 2.1

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

Answer:
It is given that
(x/3 + 1, y – 2/3) = (5/3, 1/3)
Since the ordered pairs are equal, the corresponding elements will also be equal.
Therefore,
⇒ x/3 + 1 = 5/3 and y – 2/3 = 1/3
⇒ x/3 = 5/3 – 1 and y = 1/3 + 2/3
⇒ x/3 = 2/3 and y = 3/3
⇒ x/3 = 2/3 and y = 1
⇒ x = 2 and 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:
It is given that set A has 3 elements and the elements of set B are 3, 4, and 5.
Number of elements in set B = 3
Number of elements in (A * B) = (Number of elements in A) * (Number of elements in B)
                                                       = 3 * 3 = 9
Thus, the number of elements in (A * B) is 9
 
 

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

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

Answer:
(i) False
If P = {m, n} and Q = {n, m}, then
P * Q = {(m, m), (m, n), (n, m), (n, n)}
(ii) True
(iii) True
 

Question 5:
If A = {–1, 1}, find A * A * A.

Answer:
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}
Now, 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:
It is given that A * B = {(a, x), (a, y), (b, x), (b, y)}
We know that the Cartesian product of two non-empty sets P and Q is defined as
P * Q = {(p, q): p ∈ P, q ∈ Q}
So, A is the set of all first elements and B is the set of all second elements.
Thus, 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.

Answer:
(i) To verify: A * (B ∩ C) = (A * B) ∩ (A * C)
We have B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = ф
L.H.S. = A * (B ∩ C) = A × ф = ф
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)}
R.H.S. = (A * B) ∩ (A * C) = ф
Since L.H.S. = R.H.S
Hence, A * (B ∩ C) = (A * B) ∩ (A * C)
(ii) To verify: A * C is a subset of B * D
A * C = {(1, 5), (1, 6), (2, 5), (2, 6)}
A * 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)}
We can observe that all the elements of set A * C are the elements of set B * D.
Therefore, A * C is a subset of B * D.