Question

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 $P\times Q=\left\{\right(m,n),(n,m\left)\right\}$
(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 $A\times (B\cap \varphi )=\varphi$

Answer 1

(i) Given $P=\{m,n\}$ and $Q=\{n,m\}$ then
$P\times Q=\left\{\right(m,n),(m,m),(n,n),(n,m\left)\right\}$

So, given value of $P\times Q$ is incorrect.

Hence, the given statement (i) is false.

(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 $y\in B$

Hence, the given statement (ii) is true.

(iii) $A=\{1,2\}$ and $B=\{3,4\}$

$A\times (B\cap \varphi )=A\times \varphi =\varphi$

So, (iii) is true.