Question

For three sets A, B and C, show that

(i) $A\cap B=A\cap C$ need not imply B = C.

(ii) $A\subset B\Rightarrow C-B\subset C-A$

Answer 1

(i) Let A = {1,2,3}, B = {2,4,6} and C = {2,5,7}

Then

$A\cap B$ ={2}

and $A\cap B$ ={2}

Hence, $A\cap B=A\cap C,$ but clearly B $\ne$ C.

(ii) Given $A\subset B$

To show: C- B $\subset$ C - A

Let $xϵC-B$

$\Rightarrow xϵCandx\text{/}ϵB$ [$\therefore$ By definition of C- B]

$\Rightarrow xϵCand\text{/}ϵA$ [$\therefore A\subset B$]

This can be seen by the venn diagram above

$\Rightarrow xϵC-A$ [by definition of C- A]

Thus $xϵC-B\Rightarrow xϵC-A.$ This is true for all $xϵC-B$

$\therefore C-B\subset C-A$