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
From Complement Union with Superset is Universe: Corollary:


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

and from Empty Intersection iff Subset of Complement:


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

The result follows by definition of set equality.