Characterization of Set Equals Union of Sets
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Theorem
Let $A$ be a set.
Let $\BB$ be a set of sets.
Then $A = \ds \bigcup \BB$ if and only if:
- $\forall a \in A : \exists B \in \BB : a \in B$
- $\forall B \in \BB : B \subseteq A$
Proof
Necessary Condition
Let $A = \ds \bigcup \BB$.
By definition of set union:
- $\forall a \in A = \ds \bigcup \BB : \exists B \in \BB : a \in B$
From Set is Subset of Union:
- $\forall B \in \BB : B \subseteq \ds \bigcup \BB = A$
$\Box$
Sufficient Condition
Let:
- $\forall a \in A : \exists B \in \BB : a \in B$
- $\forall B \in \BB : B \subseteq A$
From set union
- $\forall a \in A : a \in \bigcup \BB$
By definition of subset:
- $A \subseteq \bigcup \BB$
From Union of Subsets is Subset:
- $\bigcup \BB \subseteq A$
By definition of set equality:
- $A = \bigcup \BB$
$\blacksquare$