Union is Smallest Superset/Family of Sets

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Theorem

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.


Then for all sets $X$:

$\ds \paren {\forall i \in I: S_i \subseteq X} \iff \bigcup_{i \mathop \in I} S_i \subseteq X$

where $\ds \bigcup_{i \mathop \in I} S_i$ is the union of $\family {S_i}$.


Proof

Necessary Condition

From Union of Family of Subsets is Subset we have that:

$\ds \paren {\forall i \in I: S_i \subseteq X} \implies \bigcup_{i \mathop \in I} S_i \subseteq X$

$\Box$


Sufficient Condition

Now suppose that $\ds \bigcup_{i \mathop \in I} S_i \subseteq X$.

Consider any $i \in I$ and take any $x \in S_i$.

From Set is Subset of Union of Family we have that:

$\ds S_i \subseteq \bigcup_{i \mathop \in I} S_i$

Thus:

$\ds x \in \bigcup_{i \mathop \in I} S_i$

But:

$\ds \bigcup_{i \mathop \in I} S_i \subseteq X$

So it follows that $S_i \subseteq X$.


So:

$\ds \bigcup_{i \mathop \in I} S_i \subseteq X \implies \paren {\forall i \in I: S_i \subseteq X}$

$\Box$


Hence:

$\ds \paren {\forall i \in I: S_i \subseteq X} \iff \bigcup_{i \mathop \in I} S_i \subseteq X$

$\blacksquare$


Also see


Sources

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