Question

Let $U=\{-2,-1,0,1,2,3,....,10\},A=\{-2,2,3,4,5\}$ and $B=\{1,3,5,8,9\}$. Verify De Morgan's laws of complementation

Answer 1

De Morgan's laws of complementation:
i) $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
ii) $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$
$U=\{-2,-1,0,1,2,3,4,5,6,7,8,9,10\}$
$A=\{-2,2,3,4,5\}$
$B=\{1,3,5,8,9\}$
$A\cup B=\{-2,1,2,3,4,5,8,9\}$
$(A\cup B{)}^{\prime }=\{-1,0,6,7,10\}$ .... (1)
$A^{\prime }=\{-1,0,1,6,7,8,9,10\}$
$B^{\prime }=\{-2,-1,0,2,4,6,7,10\}$
$A^{\prime }\cap B^{\prime }=\{-1,0,6,7,10\}$ .... (2)
From (1) and (2), we have

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

$A\cap B=\{3,5\}$
$(A\cap B{)}^{\prime }=\{-2,-1,0,1,2,4,6,7,8,9,10\}$ .... (3)
$A^{\prime }\cup B^{\prime }=\{-2,-1,0,1,2,4,6,7,8,9,10\}$ .... (4)
From (3) and (4), we have

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