# (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

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

$\blacksquare$