# 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.

 $\displaystyle \paren {S \setminus T} \cup S$ $=$ $\displaystyle \paren {S \cap \map \complement T} \cup S$ Set Difference as Intersection with Complement $\displaystyle$ $=$ $\displaystyle S$ Union Absorbs Intersection

$\blacksquare$