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.

(i) Given: P = {m, n} and Q = {n, m}

By definition of Cartesian product of two non-empty Set P and Q:

P × Q = {(p, q): p Є P, q Є Q}

Therefore, P × Q = {(m, n), (m, m), (n, m), (n, n)}.

⇒ P × Q ≠ {(m, n), (n, m)}.

Hence, the statement is false.

(ii) Given: A and B are non-empty sets and x Є A and y Є B.

By definition of Cartesian product of two non-empty Set P and Q:

P × Q = {(p, q): p Є P, q Є Q}

⇒ A × B = {(x, y): x Є A, y Є B}

Hence, the statement is true.

(iii) Given: A = {1, 2}, B = {3, 4}

To Prove:

As

By definition if either of the two set P and Q is null set then P × Q will also be a null set. i.e. P × Q = ϕ.

⇒ .

Hence, the statement is true.