# Symmetric Difference with Empty Set

## Theorem

$S * \varnothing = S$

where $*$ denotes the symmetric difference.

## Proof

 $\displaystyle S * \varnothing$ $=$ $\displaystyle \left({S \cup \varnothing}\right) \setminus \left({S \cap \varnothing}\right)$ $\quad$ Definition of Symmetric Difference $\quad$ $\displaystyle$ $=$ $\displaystyle S \setminus \left({S \cap \varnothing}\right)$ $\quad$ Union with Empty Set $\quad$ $\displaystyle$ $=$ $\displaystyle S \setminus \varnothing$ $\quad$ Intersection with Empty Set $\quad$ $\displaystyle$ $=$ $\displaystyle S$ $\quad$ Set Difference with Empty Set is Self $\quad$

$\blacksquare$