# (A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A

## Theorem

Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$.

Then:

$\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$

where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$.

## Proof

 $\displaystyle \paren {A \cap C} \cup \paren {B \cap \map \complement C}$ $=$ $\displaystyle \O$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle A \cap C$ $=$ $\displaystyle \O$ Union is Empty iff Sets are Empty $\, \displaystyle \land \,$ $\displaystyle B \cap \map \complement C$ $=$ $\displaystyle \O$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle C$ $\subseteq$ $\displaystyle \map \complement A$ Empty Intersection iff Subset of Complement $\, \displaystyle \land \,$ $\displaystyle B$ $\subseteq$ $\displaystyle C$ Intersection with Complement is Empty iff Subset $\displaystyle \leadstoandfrom \ \$ $\displaystyle B$ $\subseteq$ $\displaystyle C \subseteq \map \complement A$

$\blacksquare$