Union of Subset of Family is Subset of Union of Family

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Theorem

Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.

Let $J \subseteq I$


Then:

$\ds \bigcup_{\alpha \mathop \in J} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} A_\alpha$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Proof

\(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in J} A_\alpha\)
\(\ds \leadsto \ \ \) \(\ds \exists \alpha \in J: \, \) \(\ds x\) \(\in\) \(\ds A_\alpha\) Definition of Union of Family
\(\ds \leadsto \ \ \) \(\ds \exists \alpha \in I: \, \) \(\ds x\) \(\in\) \(\ds A_\alpha\) Definition of Subset: $J \subseteq I$
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} A_\alpha\) Set is Subset of Union

$\blacksquare$


Also see


Sources