Set Union Preserves Subsets/Families of Sets

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:

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


Proof

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

By definition of subset:

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

$\blacksquare$


Sources

  • 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Exercise $1 \ \text{(e)}$
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