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

 (x/3)+1=5/3                  y-(2/3)=1/3

(x/3)=(5/3)-1                 y=(1/3)+(2/3)

x/3=2/3                              y=3/3

3x = 6                                   y = 1    

x = 2

2. If set A has 3 elements and the set B = {3, 4, 5} then find the number of elements in (A * B)

As A has 3 elements and the set B also has three elements then, A * B = 3 * 3 = 9

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

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)}

4. State whether each of the following statements are true or false. If the statement is false, requite the statement correctly.

(i) If P = {m, n} and Q = {n, m}, then P * Q = {(m, n), (n, m)}

False, ∵ P * Q = {(m, n), (m, m), (n, n), (n, m)}

(ii) If A & B are non-empty sets, then A * B is a non-empty set of ordered pairs (x, y) such that x Є A and y Є B

True

(iii) If A = {1, 2}, B = {3, 4} then A * (B ∩ 𝜙) = 𝜙

True.

5. If A = {1, 1}, find A * A * A

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)}

6. If A * B = {(a, x), (a, y), (b, x), (b, y)} Find A & B

A = {a, b}, B = {x, y}

7. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, D = {5, 6, 7, 8}. Verify that

(i) A * (B ∩ C) = (A * B) ∩ (A * C)

B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = 𝜙

Taking L.H.S = A * (B ∩ C) = {1, 2} * 𝜙 = 𝜙

(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)}

Taking R.H.S = (A * B) ∩ (A * C)

= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (1, 5), (1, 6), (2, 5), (2, 6)} = 𝜙

(ii) (A * C) is a subset of (B * D)

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, 6), (4, 5), (4, 7), (4, 8)}

All the elements of set A * B Є B * D

∴ (A * C) ⊂ (B * D)

8. Let A = {1, 2}, B = {3, 4}.Write A * B. How many subsets will A * B have? List them.

A * B = {(1, 3), (1, 4), (2, 3), (2, 4)}

No. of subsets of A * B = (2)⁴ = 16

Subset of A * B = 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, 4)}, {(2, 3)}, {(2, 4)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, {(2, 4)}}, {(1, 3)}, {(2, 3)}, {(1, 3)}, {(2, 4)}, {(1, 4)}

9. Let A & 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 & B, where x, y and z are distinct elements.

A = {x, y, z},                        B = {1, 2}

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.

A = {-1, 0, 1},                     A = {-1, 0, 1}

A * A = {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 0), (0, 1), (1, -1), (1, 0), (1, 1)}