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 $\Leftrightarrow$.

There are also some special terms used to indicate the presence of a necessary and sufficient condition, usually used for statements of special significance:

A criterion is a proposition that expresses a necessary and sufficient condition for a statement to be true. The term is mostly used in cases where this condition is easier to check than the statement itself.
While — in the strict sense of the word — the condition given in a criterion has to be necessary and sufficient, the term is sometimes (mostly out of tradition) also used for conditions which are only sufficient.

A characterization of a mathematical object, a class of objects, or a property, is an alternative description equivalent to a previously given definition, i.e., a necessary and sufficient condition. This term is mainly used in cases where the condition is mathematically interesting and provides new insight.

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

Examples

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   $(a_{n}),\ 0\leq a_{n}\in \mathbb {R}$   the following is true:

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

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

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

Cauchy convergence criterion

A sequence (a_n) of real numbers is convergent if and only if for all ε > 0 there is a number N such that |a_n − a_m| < ε for all n,m > N.

Root test

A series   $\sum a_{n}$   of (real or complex) numbers a_n

• is convergent if   $\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}<1$, and
• is divergent if   $\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}>1$.

This test is traditionally often referred to as a "criterion" even though

• it does not decide in the case where   $\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}=1$,

and therefore is not a true criterion.

Characterization of circles

A circle (in the plane) is usually defined as

• a curve such that its points all have the same distance from a given point.

An example of an alternative characterization is the following:

• Circles are the plane curves with non-zero constant curvature.