Chapter 2 Relations and Functions Ex 2.1 Solutions
Question - 1 : - If┬а
┬а, find the values of x and y.
Answer - 1 : -
Given,
┬а
As the ordered pairs are equal, the corresponding elements should also be equal.
Thus,
x/3 + 1 = 5/3 and y тАУ 2/3 = 1/3
Solving, we get
x + 3 = 5 and 3y тАУ 2 = 1 [Taking L.C.M and adding]
x = 2 and 3y = 3
Therefore,
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 ├Ч B)?
Answer - 2 : -
Given, set A has 3 elements and the elements of set B are {3, 4, and 5}.
So, the number of elements in set B = 3
Then, the number of elements in (A ├Ч B) = (Number of elements in A) ├Ч (Number of elements in B)
= 3 ├Ч 3 = 9
Therefore, the number of elements in (A ├Ч B) will be 9.
Question - 3 : - If G = {7, 8} and H = {5, 4, 2}, find G ├Ч H and H ├Ч G.
Answer - 3 : -
Given, G = {7, 8} and H = {5, 4, 2}
We know that,
The Cartesian product of two non-empty sets P and Q is given as
P ├Ч Q = {(p, q): p тИИ P, q тИИ Q}
So,
G ├Ч H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}
H ├Ч 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 ├Ч Q = {(m, n), (n, m)}.
(ii) If A and B are non-empty sets, then A ├Ч B is a non-empty set of ordered pairs (x, y) such that x тИИ A and y тИИ B.
(iii) If A = {1, 2}, B = {3, 4}, then A ├Ч (B тИй ╬ж) = ╬ж.
Answer - 4 : -
(i) The statement is False. The correct statement is:
If P = {m, n} and Q = {n, m}, then
P ├Ч Q = {(m, m), (m, n), (n, m), (n, n)}
(ii) True
(iii) True
Question - 5 : - If A = {тАУ1, 1}, find A ├Ч A ├Ч A.
Answer - 5 : -
The A ├Ч A ├Ч A for a non-empty set A is given by
A ├Ч A ├Ч A = {(a, b, c): a, b, c тИИ A}
Here, It is given A = {тАУ1, 1}
So,
A ├Ч A ├Ч A = {(тАУ1, тАУ1, тАУ1), (тАУ1, тАУ1, 1), (тАУ1, 1, тАУ1), (тАУ1, 1, 1), (1, тАУ1, тАУ1), (1, тАУ1, 1), (1, 1, тАУ1), (1, 1, 1)}
Question - 6 : - If A ├Ч B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.
Answer - 6 : -
Given,
A ├Ч B = {(a, x), (a, y), (b, x), (b, y)}
We know that the Cartesian product of two non-empty sets P and Q is given by:
P ├Ч Q = {(p, q): p тИИ P, q тИИ Q}
Hence, A is the set of all first elements and B is the set of all second elements.
Therefore, 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 ├Ч (B тИй C) = (A ├Ч B) тИй (A ├Ч C)
(ii) A ├Ч C is a subset of B ├Ч D
Answer - 7 : -
Given,
A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}
(i) To verify: A ├Ч (B тИй C) = (A ├Ч B) тИй (A ├Ч C)
Now, B тИй C = {1, 2, 3, 4} тИй {5, 6} = ╬ж
Thus,
L.H.S. = A ├Ч (B тИй C) = A ├Ч ╬ж = ╬ж
Next,
A ├Ч B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
A ├Ч C = {(1, 5), (1, 6), (2, 5), (2, 6)}
Thus,
R.H.S. = (A ├Ч B) тИй (A ├Ч C) = ╬ж
Therefore, L.H.S. = R.H.S
тАУ Hence verified
(ii) To verify: A ├Ч C is a subset of B ├Ч D
First,
A ├Ч C = {(1, 5), (1, 6), (2, 5), (2, 6)}
And,
B ├Ч 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)}
Now, itтАЩs clearly seen that all the elements of set A ├Ч C are the elements of set B ├Ч D.
Thus, A ├Ч C is a subset of B ├Ч D.
тАУ Hence verified
Question - 8 : - Let A = {1, 2} and B = {3, 4}. Write A ├Ч B. How many subsets will A ├Ч B have? List them.
Answer - 8 : -
Given,
A = {1, 2} and B = {3, 4}
So,
A ├Ч B = {(1, 3), (1, 4), (2, 3), (2, 4)}
Number of elements in A ├Ч B is n(A ├Ч B) = 4
We know that,
If C is a set with n(C) = m, then n[P(C)] = 2m.
Thus, the set A ├Ч B has 24 = 16 subsets.
And, these subsets are as below:
╬ж, {(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, 3), (2, 3), (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 ├Ч B, find A and B, where x, y and z are distinct elements.
Answer - 9 : -
Given,
n(A) = 3 and n(B) = 2; and (x, 1), (y, 2), (z, 1) are in A ├Ч B.
We know that,
A = Set of first elements of the ordered pair elements of A ├Ч B
B = Set of second elements of the ordered pair elements of A ├Ч B.
So, clearly x, y, and z are the elements of A; and
1 and 2 are the elements of B.
As n(A) = 3 and n(B) = 2, it is clear that set A = {x, y, z} and set B = {1, 2}.
Question - 10 : - The Cartesian product A ├Ч A has 9 elements among which are found (тАУ1, 0) and (0, 1). Find the set A and the remaining elements of A ├Ч A.
Answer - 10 : -
We know that,
If n(A) = p and n(B) = q, then n(A ├Ч B) = pq.
Also, n(A ├Ч A) = n(A) ├Ч n(A)
Given,
n(A ├Ч A) = 9
So, n(A) ├Ч n(A) = 9
Thus, n(A) = 3
Also given that, the ordered pairs (тАУ1, 0) and (0, 1) are two of the nine elements of A ├Ч A.
And, we know in A ├Ч A = {(a, a): a тИИ A}.
Thus, тАУ1, 0, and 1 has to be the elements of A.
As n(A) = 3, clearly A = {тАУ1, 0, 1}.
Hence, the remaining elements of set A ├Ч A are as follows:
(тАУ1, тАУ1), (тАУ1, 1), (0, тАУ1), (0, 0), (1, тАУ1), (1, 0), and (1, 1)