Answer
Done below.
Work Step by Step
a). Of course, if $x$ is in $A \cup \bar{A}$, then it is in $U$. For the other direction, if $x$ is in $U$, then it is either in $A$ or $\bar{A}$. Hence it is in the union.
b). Of course, the empty set is a subset of $A \cap \bar{A}$. For the other direction, if $x$ is in both $A$ and $\bar{A}$, then there is no possibly choice for $x$ and so we have the empty set.
