Intersection Subset Union

Theorem

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

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

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S \cap T\)	\(\subseteq\)	\(\displaystyle S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Intersection Subset
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S\)	\(\subseteq\)	\(\displaystyle S \cup T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Subset of Union
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle S \cap T\)	\(\subseteq\)	\(\displaystyle S \cup T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Subsets Transitive

$\blacksquare$

Sources

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