Relative Complement of Relative Complement/Proof 2

Theorem

$\relcomp S {\relcomp S T} = T$

Proof

\(\ds \relcomp S {\relcomp S T}\)

\(=\)

\(\ds S \setminus \paren {S \setminus T}\)

Definition of Relative Complement

\(\ds \)

\(=\)

\(\ds S \cap T\)

Set Difference with Set Difference

The definition of the relative complement requires that:

$T \subseteq S$

But from Intersection with Subset is Subset‎:

$T \subseteq S \iff T \cap S = T$

Thus:

$\relcomp S {\relcomp S T} = T$

follows directly.

$\blacksquare$