Empty Intersection iff Subset of Complement/Corollary/Proof 2
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Corollary to Empty Intersection iff Subset of Complement
Let $A, B, S$ be sets such that $A, B \subseteq S$.
Then:
- $\exists X \in \powerset S: \paren {A \cap X} \cup \paren {B \cap \complement_S \paren X} = \O \iff A \cap B = \O$
where $\overline X$ denotes the relative complement of $X$ in $S$.
Proof
We have:
where the universe $\Bbb U$ is posited.
Let $S$ take the position of $\Bbb U$.
Let $C = X$.
Then we have:
- $\paren {A \cap X} \cup \paren {B \cap \relcomp S X} = \O \iff B \subseteq X \subseteq \relcomp S A$
Thus we have shown that:
- $B \subseteq \relcomp S A$
and it follows from Empty Intersection iff Subset of Complement that:
- $A \cap B = \O$
$\blacksquare$