Symmetric Difference with Complement

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Theorem

The symmetric difference of a set with its complement is the universe:

$S \symdif \relcomp {} S = \mathbb U$


Proof

\(\ds S \symdif \relcomp {} S\) \(=\) \(\ds \paren {S \cup \relcomp {} S} \setminus \paren {S \cap \relcomp {} S}\) Definition 2 of Symmetric Difference
\(\ds \) \(=\) \(\ds \paren {S \cup \relcomp {} S} \setminus \O\) Intersection with Complement
\(\ds \) \(=\) \(\ds \mathbb U \setminus \O\) Union with Complement
\(\ds \) \(=\) \(\ds \mathbb U\) Set Difference with Empty Set is Self

$\blacksquare$


Sources