Question:
State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly.
(i) If $\mathrm{P}=\{m, n\}$ and $\mathrm{Q}=\{n, m\}$, then $\mathrm{P} \times \mathrm{Q}=\{(m, n),(n, m)\}$.
(ii) If $A$ and $B$ are non-empty sets, then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$.
(iii) If $A=\{1,2\}, B=\{3,4\}$, then $A \times(B \cap \Phi)=\Phi$.
Solution:
(i) False
If $\mathrm{P}=\{m, n\}$ and $\mathrm{Q}=\{n, m\}$, then
$\mathrm{P} \times \mathrm{Q}=\{(m, m),(m, n),(n, m),(n, n)\}$
(ii) True
(iii) True