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axiom of union (Axiom)

For any $ X$ there exists a set $ Y = \bigcup X$.

The Axiom of Union is an axiom of Zermelo-Fraenkel set theory. In symbols, it reads

$\displaystyle \forall X \exists Y \forall u (u \in Y \leftrightarrow \exists z (z \in X \land u \in z)). $

Notice that this means that $ Y$ is the set of elements of all elements of $ X$. More succinctly, the union of any set of sets is a set. By Extensionality, the set $ Y$ is unique. $ Y$ is called the union of $ X$.

In particular, the Axiom of Union, along with the Axiom of Pairing allows us to define

$\displaystyle X \cup Y = \bigcup \{ X, Y \}, $
as well as the triple
$\displaystyle \{ a, b, c \} = \{ a, b \} \cup \{ c \} $
and therefore the $ n$-tuple
$\displaystyle \{ a_1, \ldots, a_n \} = \{ a_1 \} \cup \cdots \cup \{ a_n \} $



"axiom of union" is owned by Sabean.
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Other names:  union
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Cross-references: axiom of pairing, extensionality, Zermelo-Fraenkel set theory, axiom
There are 14 references to this entry.

This is version 5 of axiom of union, born on 2003-06-25, modified 2003-06-26.
Object id is 4394, canonical name is AxiomOfUnion.
Accessed 13837 times total.

Classification:
AMS MSC: 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments)
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