# 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