# 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:

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

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

$$\{a,b\}=\text{the unique}c\text{such that}\forall x(x\in c\leftrightarrow x=a\vee x=b).$$ |

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

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

We may define the ordered $n$-tuple recursively

$$({a}_{1},\mathrm{\dots},{a}_{n})=(({a}_{1},\mathrm{\dots},{a}_{n-1}),{a}_{n}).$$ |

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 |