Intersection is Empty and Union is Universe if Sets are Complementary
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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.
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $11 \ \text{(c)}$