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

$U=\{1,2,3,4,5,6,7,8,9\}$
$A=\{2,4,6,8\}$
$B=\{2,3,5,7\}$
(i) $(A\cup B{)}^{\prime }={\{2,3,4,5,6,7,8\}}^{\prime }=\{1,9\}$
$A^{\prime }\cap B^{\prime }=\{1,3,5,7,9\}\cap \{1,4,6,8,9\}=\{1,9\}$
$\therefore (A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
(ii) $(A\cap B{)}^{\prime }={\left\{2\right\}}^{\prime }=\{1,3,4,5,6,7,8,9\}$
$A^{\prime }\cup B^{\prime }=\{1,3,5,7,9\}\cup \{1,4,6,8,9\}=\{1,3,4,5,6,7,8,9\}$
$\therefore (A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$