Question

Let U = {1,2,3,4,5,6,7,8,9},

A = {2,4,6,8} and B = {2,3,5,7}.

Verify that :

(i) $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

(ii) $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$.

Answer 1

i) U = {1,2,3,4,5,6,7,8,9}

A = {2,4,6,8}

B = {2,3,5,7}

We have ,

$A\cup B$ = {$x:xϵAorxϵB$}

= {2,3,4,5,6,7,8}

$\therefore (A\cup B{)}^{\prime }$ = {$xϵU:x\text{/}ϵB$}

= {1,9}

A' = {$xϵU:x\text{/}ϵA$}

= {1,3,5,7,9}

$B^{\prime }$ = {$xϵU:x\mathrm{ϵ/}B$}

{1,4,6,8,9}

Hence $A^{\prime }\cap B^{\prime }$ = {1, 9}

Hence, $(A\cup B{)}^{\prime }$= $A^{\prime }\cap B^{\prime }$ = {1,9}

(ii) $A\cap B$ = {$x:xϵAandxϵA\cup B$}

= {2}

$\therefore (A\cap B{)}^{\prime }$ = {$xϵU:x\text{/}ϵA\cap B$}

= {1,3,4,5,6,7,8,9}

Also

$A\cup B$' = {$x:xϵA^{\prime }orxϵB^{\prime }$}

= {1,3,4,5,6,7,8,9}

Hence,$(A\cup B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

= {1,3,4,5,6,7,8,9}