Let $A,B$ be sets. 1. If $A$ has an empty intersection with $B,$ then $A$ is contained in the complement of $B,$ formally $$A\cap B=\emptyset\Rightarrow A\subseteq B^C.$$ 1. If $A$ has an empty intersection with $B,$ then $A$ is contained in the complement of $B,$ formally $$A\cap B=\emptyset\Rightarrow A\subseteq B^C.$$
Proofs: 1