Intersection is Empty and Union is Universe if Sets are Complementary

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


$A \cap B = \O$ and $A \cup B = \Bbb U$

if and only if:

$B = \relcomp {\Bbb U} A$

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


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.