# 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