Axiom of Pairing

Set Theory · Axiom Academy

For any two sets, there exists a set containing exactly those two sets For any sets a and b, there exists a set that contains exactly a and b as elements. In formal notation, using the existential quantifier: Let's see how this works with concrete sets. Suppose we have: The Axiom of Pairing guarantees the existence of a set containing exactly these two sets: 3. Special Case: Pairing a Set with Itself What happens if we pair a set with itself? The axiom still applies! If a = 1, 2, 3 , then we can form the pair a, a . But by extensionality, since both elements are the same, a, a = a . This gives us a way to form singleton sets - sets containing exactly one element. The Axiom of Pairing enables several important constructions: Ordered pairs: We can define (a, b) = a , a, b Singleton sets: a is the unique set containing only a Building finite sets: Starting point for constructing any finite set Relations and functions: Foundation for defining mathematical relationships

This is the written version of the interactive lesson above. See the full Set Theory course.