Question

If A, B and C be three non empty sets given in such a way that $A\times B=A\times C$, then prove that B = C.

Answer 1

Let $A\times B=A\times C$ and we have to prove that B = C

Let $bϵB$. Then,

$bϵB\Rightarrow (a,b)ϵA\times B$ for every $aϵA$

$\Rightarrow (a,b)ϵA\times C$ for every $aϵA$ $[\because A\times B=A\times C]$

$\Rightarrow bϵC$

$\therefore B\subseteq C\dots \dots \left(i\right)$

Again, let $cϵC$. Then,

$cϵC\Rightarrow (a,c)ϵA\times C$ for every $aϵA$

$\Rightarrow (a,c)ϵA\times BforeveryaϵA$ $[\because A\times C=A\times B]$

$\Rightarrow cϵB$

$\therefore C\subseteq B\dots \dots \left(ii\right)$

From (i) and (ii), we get B = C

Hence, $A\times B=A\times C\Rightarrow B=C$