Definition:Doubleton/Class Theory

Definition

Let $a$ and $b$ be sets.

The class $\set {a, b}$ is a doubleton (class).

It is defined as the class of all $x$ such that $x = a$ or $x = b$:

$\set {a, b} = \set {x: x = a \lor x = b: a \ne b}$

Also known as

A doubleton is also known as an unordered pair, or even simply pair set or pair if no ambiguity results.

Also see

• Axiom:Axiom of Pairing (Class Theory)

• Union of Disjoint Singletons is Doubleton for a proof from the Zermelo-Fraenkel axioms that $\set a \cup \set b = \set {a, b}$ when $a \ne b$.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.1$
• 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: $\S 4$ The pairing axiom