Symmetric Difference with Empty Set

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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 $\quad$

$\blacksquare$


Sources