Necessary and sufficient

In mathematics, the phrase "necessary and sufficient" is frequently used, for instance, in the formulation of theorems, in the text of proofs when a step has to be justified, or when an alternative version for a definition is given.
To say that a statement is "necessary and sufficient" to another statement means that the statements are either both true or both false.

Another phrase with the same meaning is "if and only if" (abbreviated to "iff").

In formulae "necessary and sufficient" is denoted by $\Leftrightarrow$.

Necessary and sufficient

A statement A is

"a necessary and sufficient condition",

or shorter,

"necessary and sufficient"

for another statement B if it is both

• a necessary condition

and

• a sufficient condition

for B.

Necessary

The statement

• A is a necessary condition for B

or shorter

• A is necessary for B

means precisely the same as each of the following statements:

• If A is false then B cannot be true
• B is false whenever A does not hold
• B implies A

Sufficient

The statement

• A is a sufficient condition for B

or shorter

• A is sufficient for B

means precisely the same as each of the following statements:

• A implies B
• B holds whenever A is true

Example

For a sequence of positive real numbers to converge against a real number

• it is necessary that the sequence is bounded,
• it is sufficient that the sequence is monotone decreasing,
• it is necessary and sufficient that it is a Cauchy sequence.

The same statements are expressed by:

• For a sequence   $(a_{n}),\ 0\leq a_{n}\in {\textrm {R}}$   the following is true:

$(\exists a\in {\textrm {R}})\lim _{n\to \infty }a_{n}=a\ \Rightarrow \ (a_{n})\ {\text{is bounded}}$

$(\exists a\in {\textrm {R}})\lim _{n\to \infty }a_{n}=a\ \Leftarrow \ (a_{n})\ {\text{is monotone decreasing}}$

$(\exists a\in {\textrm {R}})\lim _{n\to \infty }a_{n}=a\ \Leftrightarrow \ (a_{n})\ {\text{is a Cauchy sequence}}$