Set Difference Union First Set is First Set

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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.

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

$\blacksquare$