Intersection is Subset of Union

Theorem

The intersection of two sets is a subset of their union:

$S \cap T \subseteq S \cup T$


Proof

\(\ds S \cap T\) \(\subseteq\) \(\ds S\) Intersection is Subset
\(\ds S\) \(\subseteq\) \(\ds S \cup T\) Set is Subset of Union
\(\ds \leadsto \ \ \) \(\ds S \cap T\) \(\subseteq\) \(\ds S \cup T\) Subset Relation is Transitive

$\blacksquare$


Sources

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.4$. Union
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