Symmetric Difference of Equal Sets

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Theorem

The symmetric difference of two equal sets is the empty set:

$S = T \iff S \symdif T = \O$


Proof

\(\ds S = T\) \(\leadstoandfrom\) \(\ds S \subseteq T \land T \subseteq S\) Definition of Set Equality
\(\ds \) \(\leadstoandfrom\) \(\ds \paren {S \setminus T = \O} \land \paren {T \setminus S = \O}\) Set Difference with Superset is Empty Set‎
\(\ds \) \(\leadstoandfrom\) \(\ds \paren {S \setminus T} \cup \paren {T \setminus S} = \O\) Union is Empty iff Sets are Empty
\(\ds \) \(\leadstoandfrom\) \(\ds S \symdif T = \O\) Definition of Symmetric Difference

$\blacksquare$


Sources