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