# Intersection is Empty and Union is Universe if Sets are Complementary

## Theorem

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

Then:

$A \cap B = \O$ and $A \cup B = \Bbb U$
$B = \relcomp {\Bbb U} A$

where $\relcomp {\Bbb U} A$ denotes the complement of $A$ with respect to $\Bbb U$.

## Proof

$A \cup B = \mathbb U \iff \relcomp {\Bbb U} A \subseteq B$
$A \cap B = \O \iff B \subseteq \relcomp {\Bbb U} A$

The result follows by definition of set equality.

$\blacksquare$