# Symmetric Difference with Universe

## Theorem

$\mathbb U \symdif S = \map \complement S$

where:

$\mathbb U$ denotes the universe
$\symdif$ denotes symmetric difference.

## Proof

 $\ds \mathbb U \symdif S$ $=$ $\ds \mathbb U \cup S \setminus \mathbb U \cap S$ Definition 2 of Symmetric Difference $\ds$ $=$ $\ds \mathbb U \cup S \setminus S$ Intersection with Universe $\ds$ $=$ $\ds \mathbb U \setminus S$ Union with Universe $\ds$ $=$ $\ds \map \complement S$ Definition of Set Complement

$\blacksquare$