Question

Fill in the blanks to make each of the following a true statement:
$\left(i\right)A\cup A^{\prime }=....\left(ii\right){\varphi }^{\prime }\cap A=....$
$\left(iii\right)A\cap A^{\prime }=....\left(iv\right)U^{\prime }\cap A=....$

Answer 1

$\left(i\right)$ We know, from Venn-diagram $A\cup A^{\prime }=U$



$\left(ii\right)$ We know, ${\varphi }^{\prime }=U-\varphi$
$\Rightarrow {\varphi }^{\prime }=U$
So, ${\varphi }^{\prime }\cap A=U\cap A=A$



$\left(iii\right)$ From the Venn-diagram we know, $A\cap A^{\prime }=\varphi$


$\left(iv\right)U^{\prime }=U-U=\varphi$
So, $U^{\prime }\cap A=\varphi \cap A=\varphi$