Characterization of Set Equals Union of Sets

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

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:

$A = \bigcup \BB$

$\blacksquare$