Question

Suppose $U=\{1,2,3,4,5,6,7,8,9\},A=\{1,2,3,4\}$ and $B=\{2,4,6,8\}$. Write down the following sets:
$A^{\prime }$. How $(A\cup B{)}^{\prime }$ is related to $A^{\prime }$ and $B^{\prime }$? What relation you see between $(A\cap B{)}^{\prime }$ and $A^{\prime }$ and $B^{\prime }$?

Answer 1

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

$A=\{1,2,3,4\}$

$B=\{2,4,6,8\}$

$A\cup B=\{1,2,3,4,6,8\}$

$(A\cup B{)}^{\prime }=\{5,7,9\}$

$A^{\prime }=\{5,6,7,8,9\}$

$B^{\prime }=\{1,3,5,7,9\}$

$A^{\prime }\cap B^{\prime }=\{5,7,9\}$

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

$A\cap B=\{2,4\}$

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

$A^{\prime }\cup B^{\prime }=\{1,2,3,4,5,6,7,8,9\}$

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