Axiom of pairing

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by Zermelo (1908) as a special case of his axiom of elementary sets.

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

or in words:

or in simpler words:

What the axiom is really saying is that, given two sets A and B, we can find a set C whose members are exactly A and B. We can use the axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is:

{A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair.

The axiom of pairing also allows for the definition of ordered pairs. For any sets ${\displaystyle a}$ and ${\displaystyle b}$, the ordered pair is defined by the following:

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