axiom of pairing

For any a and b there exists a set {a,b} that contains exactly a and b.

The Axiom of Pairing is one of the axioms of Zermelo-Fraenkel set theoryMathworldPlanetmath. In symbols, it reads:


Using the Axiom of ExtensionalityMathworldPlanetmath, we see that the set c is unique, so it makes sense to define the pair

{a,b}= the unique c such that x(xcx=ax=b).

Using the Axiom of Pairing, we may define, for any set a, the singleton


We may also define, for any set a and b, the ordered pairMathworldPlanetmath


Note that this definition satisfies the condition

(a,b)=(c,d) iff a=c and b=d.

We may define the ordered n-tuple recursively

Title axiom of pairing
Canonical name AxiomOfPairing
Date of creation 2013-03-22 13:42:43
Last modified on 2013-03-22 13:42:43
Owner Sabean (2546)
Last modified by Sabean (2546)
Numerical id 7
Author Sabean (2546)
Entry type Axiom
Classification msc 03E30
Synonym pairing