Relative Complement inverts Subsets of Relative Complement
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Theorem
Let $S$ be a set.
Let $A \subseteq S, B \subseteq S$ be subsets of $S$.
Then:
- $A \subseteq \relcomp S B \iff B \subseteq \relcomp S A$
where $\complement_S$ denotes the complement relative to $S$.
Proof
We have:
\(\ds A \subseteq \relcomp S B\) | \(\iff\) | \(\ds \relcomp S {\relcomp S B} \subseteq \relcomp S A\) | Relative Complement inverts Subsets | |||||||||||
\(\ds \) | \(\iff\) | \(\ds B \subseteq \relcomp S A\) | Relative Complement of Relative Complement |
$\blacksquare$