Question

For any two sets A and B , prove that :

$A\cap B=\varphi \Rightarrow A\subseteq B^{\prime }$.

Answer 1

Given $A\cap B=\varphi$, i.e A and B are disjoint sets this can represented by venn diagram as follows

To shows : $A\subseteq B^{\prime }$

This is clear from the venn diagram itself

$\because$ A is lying in the complement of B, but we give a proof of it.

S let $xϵA$

$\because A\cap B=\varphi$

$\therefore x\text{/}ϵB$

and so $xϵB^{\prime }$ [$\therefore x\text{/}ϵ\Rightarrow xϵB^{\prime }$]

Thus $xϵA\Rightarrow xϵB^{\prime }$. This is true for all $xϵA$.

Hence ,$A\subseteq B^{\prime }$.