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 $Q$

- If an element is in $P$, it
- $P$ 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 $Q$

- An element
- $Q$ 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

- If $P$ is true, then $Q$
- $P$ 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

- $P$ can
- $Q$ is
**not**a sufficient condition for $P$- If $Q$ is true, $P$ may be true or false