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