Set is Subset of Union/Family of Sets/Proof 1
Jump to navigation
Jump to search
Theorem
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Then:
- $\ds \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$
where $\ds \bigcup_{\alpha \mathop \in I} S_\alpha$ is the union of $\family {S_\alpha}$.
Proof
Let $x \in S_\beta$ for some $\beta \in I$.
Then:
\(\ds x\) | \(\in\) | \(\ds S_\beta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \set {x: \exists \alpha \in I: x \in S_\alpha}\) | Definition of Indexed Family of Sets | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \bigcup_{\alpha \mathop \in I} S_\alpha\) | Definition of Union of Family | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds S_\beta\) | \(\subseteq\) | \(\ds \bigcup_{\alpha \mathop \in I} S_\alpha\) | Definition of Subset |
As $\beta$ was arbitrary, it follows that:
- $\ds \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$
$\blacksquare$