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Homenecessary and sufficient

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# necessary and sufficient

The statement “$p$ is sufficient for $q$” means “$p$ implies $q$”.

The statement “$p$ is necessary and sufficent for $q$” means “$p$ if and only if $q$”.

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.

Defines:

necessary, necessity, sufficient, sufficiency

Related:

UniversalAssumption, SufficientConditionOfPolynomialCongruence

Major Section:

Reference

Type of Math Object:

Definition

Parent:

## Mathematics Subject Classification

03B05*no label found*03F07

*no label found*

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## Comments

## confused

I added this entry because the words ``necessary'' and ``sufficient'' seem to always confuse me. I would not be surprised if I still have it confused and that this entry is incorrect. :-) Anyways, I wanted to put this entry here so that I have an easy reference as to which means which instead of trying to keep it straight in my head and constantly mixing it up.