Set Union Preserves Subsets/Families of Sets

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $I$ be an indexing set.

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

Let:

$\forall \beta \in I: A_\beta \subseteq B_\beta$


Then:

$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Definition of Union of Family
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle B_\alpha\) Definition of Subset
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} B_\alpha\) Definition of Union of Family

By definition of subset:

$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$

$\blacksquare$


Sources