Image of Relative Complement under Bijection is Relative Complement of Image

Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a bijection.

Let $H \subseteq S$.

Let $f \sqbrk H = K$ be the image of $H$ under $f$.

Let $\relcomp S H$ denote the relative complement of $H$ in $S$.

Then:

$f \sqbrk {\relcomp S H} = \relcomp T K$

Proof

From Set with Relative Complement forms Partition, $\set {H \mid \relcomp S H}$ forms a partition of $S$.

The result follows from Bijection Preserves Set Partition.

$\blacksquare$