Intersection is Subset of Union

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Theorem

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

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


Proof

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

$\blacksquare$


Sources