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 universal set.
\(\ds \paren {S \setminus T} \cup S\) | \(=\) | \(\ds \paren {S \cap \map \complement T} \cup S\) | Set Difference as Intersection with Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds S\) | Union Absorbs Intersection |
$\blacksquare$