Question:
State whether each of the following statements are true or false. If the statements is false, re-write the given statements 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 ∈ B and y ∈ A.
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ ϕ) = ϕ.
Solution:
(i) False
Correct statement:
If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (m, m), (n, n), (n, m)}.
(ii) False
Correct statement:
If A and B are non-empty sets, then A × B is a non-empty set of an ordered pair (x, y) such that x ∈ A and y ∈ B.
(iii) True
A = {1, 2} and B = {3, 4}
Now,
(B ∩ ϕ) = ϕ
The Cartesian product of any set and an empty set is an empty set.
∴ A × (B ∩ ϕ) = ϕ