# necessary and sufficient

The statement “$p$ is necessary for $q$” “$q$ implies (http://planetmath.org/Implication^{}) $p$”.

The statement “$p$ is sufficient for $q$” “$p$ implies (http://planetmath.org/Implication) $q$”.

The statement “$p$ is necessary and sufficent for $q$” “$p$ if and only if (http://planetmath.org/Iff) $q$”.

For an example of how these terms are used in mathematics, see the entry on complete ultrametric fields.

Biconditional^{} statements are often proven by breaking them into two implications and proving them separately. Often, the terms *necessity* and *sufficiency* are used to indicate which implication is being proven. For an example of this usage, see the entry called relationship between totatives and divisors.

Title | necessary and sufficient |

Canonical name | NecessaryAndSufficient |

Date of creation | 2013-03-22 16:07:31 |

Last modified on | 2013-03-22 16:07:31 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 12 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 03B05 |

Classification | msc 03F07 |

Related topic | UniversalAssumption |

Related topic | SufficientConditionOfPolynomialCongruence |

Defines | necessary |

Defines | necessity |

Defines | sufficient |

Defines | sufficiency |