Symmetric Difference with Empty Set

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Theorem

$S * \O = S$

where $*$ denotes the symmetric difference.


Proof

\(\displaystyle S * \O\) \(=\) \(\displaystyle \paren {S \cup \O} \setminus \paren {S \cap \O}\) Definition 2 of Symmetric Difference
\(\displaystyle \) \(=\) \(\displaystyle S \setminus \paren {S \cap \O}\) Union with Empty Set
\(\displaystyle \) \(=\) \(\displaystyle S \setminus \O\) Intersection with Empty Set
\(\displaystyle \) \(=\) \(\displaystyle S\) Set Difference with Empty Set is Self

$\blacksquare$


Sources