Question

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
$\left(i\right)$ If $P=\{m,n\}$ and $Q=\{n,m\}$, then $P\times Q=\left\{\right(m,n),(n,m\left)\right\}$
$\left(ii\right)$ 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 $y\in B$
$\left(iii\right)A=\{1,2\},B=\{3,4\}$, then $A\times (B\cap \varphi )=\varphi$

Answer 1

Given: $P=$ {$m,n$} and $Q=$ {$n,m$}
The cartesian product of two non-empty sets $A$ and $B$ is given as $A\times B=\left\{\right(a,b):a\in A,b\in B\}$
So, $P\times Q=\left\{\right(m,n),(m,m),(n,m),(n,n\left)\right\}$
Given $P\times Q$ doesn't match.
Hence, given statement is false.
Correct statemnet is: If $P=\{m,n\}$ and $Q=\{n,m\}$, then $P\times Q=\left\{\right(m,n),(m,m),(n,m),(n,n\left)\right\}$

Assume sets
Let, $A=\left\{x\right\}$ and $B=\left\{y\right\}$

Cartesian product of two sets
The cartesian product of two non-empty sets $P$ and $Q$ is given as $P\times Q=\left\{\right(p,q):p\in P,q\in Q\}$
So, $A\times B=\left\{\right(x,y\left)\right\}$
Where $x\in A$ and $y\in B$
The given statement is True.

Given: $A=\{1,2\},B=\{3,4\}$ and $A\times (B\cap \varphi )=\varphi$

Intersection of two sets
As we know, an empty set has no elements i.e. $\varphi =\left\{\right\}$ and empty set is a proper subset of any non-empty set.
So, $B\cap \varphi =\varphi$

Cartesian product of two sets
$A\times (B\cap \varphi )=A\times \varphi =\varphi$
Hence, given statement is true.