# 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 theory. In symbols, it reads:

 $\forall a\forall b\exists c\forall x(x\in c\leftrightarrow x=a\lor x=b).$

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

 $\{a,b\}=\mbox{ the unique }c\mbox{ such that }\forall x(x\in c\leftrightarrow x% =a\lor x=b).$

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

 $\{a\}=\{a,a\}.$

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

 $(a,b)=\{\{a\},\{a,b\}\}.$

Note that this definition satisfies the condition

 $(a,b)=(c,d)\mbox{ iff }a=c\mbox{ and }b=d.$

We may define the ordered $n$-tuple recursively

 $(a_{1},\ldots,a_{n})=((a_{1},\ldots,a_{n-1}),a_{n}).$
Title axiom of pairing AxiomOfPairing 2013-03-22 13:42:43 2013-03-22 13:42:43 Sabean (2546) Sabean (2546) 7 Sabean (2546) Axiom msc 03E30 pairing