---
tag: 09BJQQB3VS
kind: concept
name: Necessary and sufficient conditions
aliases: [necessary-condition, sufficient-condition]
msc: [03B05]
uses: [0ZNX6DQ3EJ, 0V6GAWTBYW]
sources:
  - url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
    locator: lead section
same_as:
  wikidata: Q875267
  wikipedia: Necessity and sufficiency
---

## Summary

When the [[implication|0ZNX6DQ3EJ]] $p \to q$ 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|0V6GAWTBYW]].
For an integer $n$, "$n$ is divisible by $4$" is sufficient but not necessary for "$n$ is even".

## Explanations

### L1 human

A sufficient condition is enough to get the result.
A necessary condition is one that you cannot do without.
To have a driving licence is necessary to drive legally, but it is not sufficient: you also need a car.