Symmetric Difference is Subset of Union

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
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