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
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \alpha$