Symmetric Difference is Subset of Union
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Theorem
The symmetric difference of two sets is a subset of their union:
- $S \symdif T \subseteq S \cup T$
Proof
\(\ds S \symdif T\) | \(=\) | \(\ds \paren {S \cup T} \setminus \paren {S \cap T}\) | Definition 2 of Symmetric Difference | |||||||||||
\(\ds \) | \(\subseteq\) | \(\ds \paren {S \cup T}\) | Set Difference is Subset |
$\blacksquare$
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets