Necessary vs. Sufficient

Daniel Weibel
Created 29 Oct 2013
Last updated 31 Oct 2017

These are some reminders for the intuition of the terms necessary and sufficient conditions.

Sets

Consider the following Venn diagram: Sufficient

• $P$ is a sufficient condition for $Q$
• If an element is in $P$, it must also be in $Q$
• $P$ is not a necessary condition for $Q$
• An element may be in $Q$ without being in $P$

Necessary

• $Q$ is a necessary condition for $P$
• An element cannot be in $P$ without also being in $Q$
• $Q$ is not a sufficient condition for $P$
• If an element is in $Q$, it may or may not be in $P$

Logic

Consider the following propositional logic formula:

$P \rightarrow Q$

Sufficient

• $P$ is a sufficient condition for $Q$
• If $P$ is true, then $Q$ must be true
• $P$ is not a necessary condition for $Q$
• $Q$ may be true if $P$ is false

Necessary

• $Q$ is a necessary condition for $P$.
• $P$ can only be true, if $Q$ is also true
• $Q$ is not a sufficient condition for $P$
• If $Q$ is true, $P$ may be true or false