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

$\blacksquare$