Symmetric Difference is Commutative

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Theorem

Symmetric difference is commutative:

$S * T = T * S$


Proof

\(\displaystyle S * T\) \(=\) \(\displaystyle \paren {S \setminus T} \cup \paren {T \setminus S}\) Definition of Symmetric Difference
\(\displaystyle \) \(=\) \(\displaystyle \paren {T \setminus S} \cup \paren {S \setminus T}\) Union is Commutative
\(\displaystyle \) \(=\) \(\displaystyle T * S\) Definition of Symmetric Difference

$\blacksquare$


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