Mathematical logic and foundations Concept

Necessary and sufficient conditions

Summary

When the implication $p \to q$ is true, $p$ is a sufficient condition for $q$ , and $q$ is a necessary condition for $p$ . A condition $p$ is necessary and sufficient for $q$ when $p \to q$ and $q \to p$ are both true, that is, when $p \leftrightarrow q$ is true.

Remarks

" $p$ only if $q$ " means $p \to q$ , and " $p$ if $q$ " means $q \to p$ . "Necessary and sufficient" is another way to say "if and only if". See biconditional. For an integer $n$ , " $n$ is divisible by $4$ " is sufficient but not necessary for " $n$ is even".

Reading path

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Proposition concept
Conjunction definition
Implication definition
Biconditional definition
Necessary and sufficient conditions concept

Tag	`09BJQQB3VS`
URI	`m2://09BJQQB3VS`
Aliases	`necessary-condition`, `sufficient-condition`
Kind	concept
Writing	W0 stub
MSC	03B05
Uses	Implication, Biconditional
Same as	Wikidata Q875267 Wikipedia Necessity and sufficiency

Necessary and sufficient conditions

Summary

Remarks

Reading path

Further reading

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