Axiom of pairing

The axiom of pairing states that for any two sets, a set exists whose members are exactly those sets. It permits singleton sets when the two sets are equal.

Unordered pair: A set whose members are two specified objects, with no order or distinction between their positions. The axiom guarantees this set exists for any two sets.
Set theory: The mathematical study of sets, membership, and the structures built from them. Pairing is one of the basic existence principles in this framework.
Axiom of union: An axiom asserting that the union of the members of a set is itself a set. Union and pairing are complementary existence principles for building sets.
Axiom of empty set: An axiom asserting that a set with no members exists. It guarantees a set with no members, unlike pairing's prescribed members.
Singleton set: A set containing exactly one member. Applying pairing to the same set twice yields a singleton.
Set: A collection treated as a single mathematical object, whose members are specified by membership. The axiom applies to sets and asserts the existence of another set.
Axiom of power set: An axiom asserting that the collection of all subsets of a set forms a set. It supplies a much larger set of possibilities than pairing's two-member construction.
Axiom of regularity: An axiom forbidding certain forms of membership loops and infinitely descending membership chains. It restricts membership structure rather than asserting a particular set's existence.
Set-builder notation: A notation that describes a set by specifying a condition its members satisfy. It can describe the pair whose existence the axiom guarantees.
Membership relation: The relation that states whether an object is an element of a set. The axiom specifies the new set through exactly which objects belong to it.

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