Set Difference Union First Set is First Set

Theorem

The union of a set difference with the first set is the set itself:

Let $S, T$ be sets.

Then:

$\paren {S \setminus T} \cup S = S$

Proof

Consider $S, T \subseteq \mathbb U$, where $\mathbb U$ is considered as the universe.

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

\(=\)

\(\ds \paren {S \cap \map \complement T} \cup S\)

Set Difference as Intersection with Complement

\(\ds \)

\(=\)

\(\ds S\)

Union Absorbs Intersection

$\blacksquare$