# Necessary and sufficient condition meaning

Suppose A is a necessary condition for B. Does this mean $A\Rightarrow B$ or does it mean $B\Rightarrow A$?

Suppose P is a sufficient condition for Q. Does this mean $P\Rightarrow Q$ or $Q\Rightarrow P$?

• – Robert Z Dec 11 '16 at 11:54

$A$ is a necessary condition for $B$ iff $B \implies A$. In other words, it cannot be true that $B$ holds while $A$ doesn't.

$P$ is a sufficient condition for $Q$ iff $P \implies Q$. In other words, as soon as $P$ holds, we know that $Q$ holds as well.

Consider a statement

"The road is wet because it has rained ."

Here rain is sufficient condition for road to be wet, but not the necessary condition , because road could be wet by any other operation , i.e. throwing water buckets on it .

Similarly consider a statement

"An expression is correct for all $n\in \mathbb N$ if it could be proved by induction."

Here proof by induction is sufficient condition for an expression to be true but it doesn't implies that if we are not able to prove it by induction , then it is false.

Hope it will help.

• You have got some good examples. +1 – Vidyanshu Mishra Dec 11 '16 at 12:17