Union is Smallest Superset
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Theorem
Let $S_1$ and $S_2$ be sets.
Then $S_1 \cup S_2$ is the smallest set containing both $S_1$ and $S_2$.
That is:
- $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \iff \paren {S_1 \cup S_2} \subseteq T$
Set of Sets
Let $T$ be a set.
Let $\mathbb S$ be a set of sets.
Then:
- $\ds \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$
General Result
Let $S$ and $T$ be sets.
Let $\powerset S$ denote the power set of $S$.
Let $\mathbb S$ be a subset of $\powerset S$.
Then:
- $\ds \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$
Family of Sets
In the context of a family of sets, the result can be presented as follows:
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 Subsets is Subset:
- $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \implies \paren {S_1 \cup S_2} \subseteq T$
$\Box$
Sufficient Condition
Let $\paren {S_1 \cup S_2} \subseteq T$:
| \(\ds S_1\) | \(\subseteq\) | \(\ds S_1 \cup S_2\) | Set is Subset of Union | |||||||||||
| \(\ds \paren {S_1 \cup S_2}\) | \(\subseteq\) | \(\ds T\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds S_1\) | \(\subseteq\) | \(\ds T\) | Subset Relation is Transitive |
Similarly for $S_2$:
| \(\ds S_2\) | \(\subseteq\) | \(\ds S_1 \cup S_2\) | Set is Subset of Union | |||||||||||
| \(\ds \paren {S_1 \cup S_2}\) | \(\subseteq\) | \(\ds T\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds S_2\) | \(\subseteq\) | \(\ds T\) | Subset Relation is Transitive |
That is:
- $\paren {S_1 \cup S_2} \subseteq T \implies \paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T}$
$\Box$
Thus from the definition of equivalence:
- $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \iff \paren {S_1 \cup S_2} \subseteq T$
$\blacksquare$