TOPICS
Exercise - 2.1

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

Solution :-
   In the Ordered Pairs, the corresponding values are equal.
   So, x/3 + 1 = 5/3
           x/3 = 5/3 - 1 
           x/3 = (5 - 3)/3
            3x = 2/3
             x = 2/3 x 3
             x = 2  
       y - 2/3 = 1/3
           y = 1/3 + 2/3
           y = (1 + 2)/3
           y = 3/3
           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 :-
   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 
   

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

Solution :-
   Given that G = {7, 8} and H = {5, 4, 2}.
   G x H = {7, 8} x {5, 4, 2}
         = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 3)}
   H x G = {5, 4, 2} x {7, 8}
         = {(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.

Solution :-
(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.

Solution :-
   Given that A = {–1, 1}.
   A x A = {–1, 1} x {–1, 1} = {(-1, -1), (-1, 1), (1, -1), (1, 1)}
   A x A x A = {(-1, -1), (-1, 1), (1, -1), (1, 1)} x {-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)}
   

Question-6 :-  If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.

Solution :-
   Given that A × B = {(a, x), (a, y), (b, x), (b, y)}.
   A = {a, b}
   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 :-
    Given that A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}.
(i) A × (B ∩ C) = (A × B) ∩ (A × C) 
    L.H.S 
  = B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = Φ  
  = A × (B ∩ C) = A × Φ = Φ 
    R.H.S    
    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)} 
  = (A × B) ∩ (A × C) = Φ 
    L.H.S. = R.H.S 
    Hence, A × (B ∩ C) = (A × B) ∩ (A × C) 
    
(ii) A × C is a subset of B × D 
    L.H.S 
    A × C = {(1, 5), (1, 6), (2, 5), (2, 6)} 
    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)} 
    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.
   

Question-8 :-  Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.

Solution :-
   Given that A = {1, 2} and B = {3, 4}.
   A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}
   n(A × B) = 4
   We know that if C is a set with n(C) = m, then n[P(C)] = 2ᵐ. 
   Therefore, the set A × B has 2⁴ = 16 subsets. 
   These 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, 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 × B, find A and B, where x, y and z are distinct elements.

Solution :-
   Given that  n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B.
   A = Set of first elements of the ordered pair elements of A × B 
   B = Set of second elements of the ordered pair elements of A × B.
   x, y, and z are the elements of A; and 1 and 2 are the elements of B. 
   Since n(A) = 3 and n(B) = 2, it is clear that 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 :-
   We know that if n(A) = p and n(B) = q, then n(A × B) = pq. ∴ n(A × A) = n(A) × n(A)
   Given that n(A × A) = 9 
   n(A) × n(A) = 9 = n(A) = 3
   The ordered pairs (–1, 0) and (0, 1) are two of the nine elements of A × A.
   We know that A × A = {(a, a): a ∈ A}. 
   Therefore, –1, 0, and 1 are elements of A. 
   Since n(A) = 3, it is clear that A = {–1, 0, 1}.
   A × A = {(–1, –1), (–1, 1), (0, –1), (0, 0), (1, –1), (1, 0), (1, 1)}.
   
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