# Set Difference Union First Set is First Set

Jump to navigation
Jump to search

## 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 |

$\blacksquare$