# Union of Subset of Family is Subset of Union of Family

Jump to navigation Jump to search

## 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 of Family

$\blacksquare$