Question

If 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

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}

(i) $A\cup B$ = {2, 4, 6, 8} $\cup$ {2, 3, 5, 7}

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

$\therefore (A\cup B{)}^{\prime }=U-(A\cup B)$

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

= {1, 9} .........(i)

$A^{\prime }$ = U - A

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

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

$B^{\prime }$ = U - B

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

= {1, 4, 6, 8, 9}

$A^{\prime }\cap B^{\prime }$ = {1, 3, 5, 7, 9} $\cap$ {1, 4, 6, 8, 9}

= {1, 9} .......(ii)

From (i) and (ii), we have

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

(ii) $A\cap B$ = {2, 4, 6, 8} $\cap$ {2, 3, 5, 7} = {2}

$(A\cap B{)}^{\prime }=U-(A\cap B)$

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

= {1, 3, 4, 5, 6, 7, 8, 9} .........(i)

$A^{\prime }$ = U - A

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

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

$B^{\prime }$ = U - B

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

= {1, 4, 6, 8, 9}

$A^{\prime }\cup B^{\prime }$ = {1, 3, 5, 7, 9} $\cup$ {1, 4, 6, 8, 9}

= {1, 3, 4, 5, 6, 7, 8, 9} .......(ii)

From (i) and (ii), we have

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