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
| \(\ds S \cap T\) | \(\subseteq\) | \(\ds S\) | 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