Equal Relative Complements iff Equal Subsets

Theorem

Let $S$ be a set.

Let $A, B \subseteq S$ be subsets of $S$.

Then:

$\relcomp S B = \relcomp S A \iff A = B$

where $\complement_S$ denotes the complement relative to $S$.

Proof

We have:

\(\ds \relcomp S B = \relcomp S A\)

\(\iff\)

\(\ds \relcomp S B \subseteq \relcomp S A \land \relcomp S A \subseteq \relcomp S B\)

Definition of Set Equivalence

\(\ds \)

\(\iff\)

\(\ds A \subseteq B \land B \subseteq A\)

Relative Complement inverts Subsets

\(\ds \)

\(\iff\)

\(\ds A = B\)

Definition of Set Equivalence

$\blacksquare$