The statement “ is necessary for ” means “ implies ”.

The statement “ is sufficient for ” means “ implies ”.

The statement “ is necessary and sufficent for ” means “ if and only if ”.

For an example of how these terms are used in mathematics, see the entry on complete ultrametric fields.

Biconditional statements are often proven by breaking them into two implications and proving them separately. Often, the terms necessity and sufficiency are used to indicate which implication is being proven. For an example of this usage, see the entry called relationship between totatives and divisors.