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".

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.