The Following Is an Outline of a Proof That $( A \cup B ) ^ { c } \subseteq A ^ { c } \cap B ^ { c }$

Essay

The following is an outline of a proof that $( A \cup B ) ^ { c } \subseteq A ^ { c } \cap B ^ { c }$ . Fill in the blanks.
Given sets $A$ and $B$ , to prove that $( A \cup B ) ^ { c } \subseteq A ^ { c } \cap B ^ { c }$ , we suppose $x \in$ then we show that $x \in \ldots$ . So suppose that Then by definition of complement, So by definition of union, it is not the case that ( $x$ is in $A$ or $x$ is in $B$ ). Consequently, $x$ is not in $A$ $x$ is not in $B$ because of De Morgan's law of logic. In symbols, this says that $x \notin A$ and $x \notin B$ . So by definition of complement, $x \in \ldots$ and $x \in$ Thus, by definition of intersection, $x \in$ [as was to be shown].