Symmetric Difference of Complements

From ProofWiki
Jump to navigation Jump to search

Theorem

The symmetric difference of two sets equals the symmetric difference of their complements:

$\map \complement S * \map \complement T = S * T$


Proof

\(\displaystyle \map \complement S * \map \complement T\) \(=\) \(\displaystyle \paren {\map \complement S \setminus \map \complement T} \cup \paren {\map \complement T \setminus \map \complement S}\) Definition of Symmetric Difference
\(\displaystyle \) \(=\) \(\displaystyle \paren {T \setminus S} \cup \paren {S \setminus T}\) Set Difference of Complements
\(\displaystyle \) \(=\) \(\displaystyle S * T\) Definition of Symmetric Difference

$\blacksquare$


Sources