# What is the difference between necessary and sufficient conditions?

• If $\quad p \implies q\quad$ ($p$ implies $q$), then $p$ is a sufficient condition for $q$.

• If $\quad \bar p \implies \bar q \quad$ (not $p$ implies not $q$), then $p$ is a necessary condition for $q$.

I don't understand what sufficient and necessary mean in this case. How do you know which one is necessary and which one is sufficient?

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Those are the definitions of necessary and of sufficient. –  Zhen Lin Dec 11 '12 at 9:37
Suppose first that $p$ implies $q$. Then knowing that $p$ is true is sufficient (i.e., enough evidence) for you to conclude that $q$ is true. It’s possible that $q$ could be true even if $p$ weren’t, but having $p$ true ensures that $q$ is also true.
Now suppose that $\text{not-}p$ implies $\text{not-}q$. If you know that $p$ is false, i.e., that $\text{not-}p$ is true, then you know that $\text{not-}q$ is true, i.e., that $q$ is false. Thus, in order for $q$ to be true, $p$ must be true: without that, you automatically get that $q$ is false. In other words, in order for $q$ to be true, it’s necessary that $p$ be true; you can’t have $q$ true while $p$ is false.
@cloud9resident: You don’t know. Whether $p$ is true or not is a completely separate issue from whether it is a necessary or sufficient condition for $q$. The latter does not depend in any way on the truth or falsity of $p$. –  Brian M. Scott Dec 12 '12 at 8:46