Does $A\cap B =\varnothing \Rightarrow B\subseteq \overline{A}$? How to prove $A\cap  B =\varnothing \Rightarrow B\subseteq \overline{A}$?
If I going by definitions, there is no $x$ s.t  $x\in A$ and $x\in B$.
But, what do we can tell about $\overline{A}$? What i'm missing? 

Let's assume  $A\cap B =\varnothing$ (start hypothesis)
Let $x \in B$
Since $A$ and $B$ are disjoint (start hypothesis), then $x \notin A$
By definition of  $\overline A$, since  $x \notin A$ then $x \in \overline A ~~~~(= \Omega - A)$
Therefore  $B\subseteq \overline A$, because for all $x \in B$, we have $x \in \overline A$

Note that the reciprocal is also true.
