# 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

 $\displaystyle S \cap T$ $\subseteq$ $\displaystyle S$ $\quad$ 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 $\quad$

$\blacksquare$