Symmetric Difference with Empty Set

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Theorem

$S * \O = S$

where $*$ denotes the symmetric difference.


Proof

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

$\blacksquare$


Sources