State whether each of the following statements are true of false. If the statement is false, re - write the given statement correctly:
i. If P = [m, n} and Q = {n, m} then P x Q = {(m, n), (n, m)}
ii. If A and B are non - empty sets, then A x 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 x (B ∩ φ) = φ.
(i) False
given: P = {m, n} and Q = {n, m}, then
P × Q = {(m, n), (m, m), (n, n), (n, m)}.
(ii) False
given: 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
given: A = {1, 2} and B = {3, 4}
∅ is represents null setand intersection of any set with null set gives null set as null set has no elements
(B ∩ ∅) = ∅
The Cartesian product of any set and an empty set will be an empty set.
∴ A × (B ∩ ∅) = ∅