# Axiom:Axiom of Unions/Set Theory

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## Axiom

For every set of sets $A$, there exists a set $x$ (the **union set**) that contains all and only those elements that belong to at least one of the sets in the $A$:

- $\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$

## Also known as

The **axiom of unions** is in fact most frequently found with the name **axiom of union**.

However, in some treatments of axiomatic set theory and class theory, for example Morse-Kelley set theory this name is used to mean something different.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ specifically uses the plural form **axiom of unions** for this, and reserves the singular form **axiom of union** for that.

Other terms that can be found to refer to the **axiom of unions**:

- the
**axiom of the sum set** - the
**axiom of amalgamation** - the
**union axiom**.

## Also see

- Results about
**the axiom of unions**can be found**here**.

## Sources

- 1955: John L. Kelley:
*General Topology*: Appendix: Functions: Axiom $\text{VI}$ - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 4$: Unions and Intersections - 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF4}$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Note $1$

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html - Weisstein, Eric W. "Axiom of the Sum Set." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomoftheSumSet.html