# Well-defined

(Redirected from Defined and undefined)
For other uses, see Definition (disambiguation).

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous.[1] A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus: not a function).[2] The term well-defined is also used to indicate whether a logical statement is unambiguous.

## Well-defined functions

Let $A_1,A_2$ be sets, let $A = A_1 \bigcup A_2$ and define $f: A \rightarrow \{1,2\}$ as $f(a)=1$ if $a \in A_1$ and $f(a)=2$ if $a \in A_2$. Then $f$ is well-defined if $A_1 \bigcap A_2 = \emptyset$. If however $A_1 \bigcap A_2 \neq \emptyset$ then $f$ is not well-defined because $f(a)$ is ambiguous for $a \in A_1 \bigcap A_2$.

In group theory, the term well-defined is often used when dealing with cosets, where a function f on a quotient group may be defined in terms of a coset representative. Here, a necessary requirement for f to be considered a function is that the output must be independent of which coset representative is chosen. The phrase f is well-defined is used to indicate that this requirement has been verified.

For example, consider $\mathbb{Z}/2\mathbb{Z}$, the integers modulo 2. Since 4 and 6 are congruent modulo 2, a function f whose domain is $\mathbb{Z}/2\mathbb{Z}$ must give the same output when the input is represented by 4 that it gives when the input is represented by 6.

A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, then f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined. It is; 0 is simply not in the domain of the function.

### Operations

In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case one can view the operation as a function of two variables and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition.

$[a]\oplus[b] = [a+b]$

The fact that this is well-defined follows from the fact that we can write any representative of $[a]$ as $a+kn$, where k is an integer. Therefore,

$[a+kn]\oplus[b] = [(a+kn)+b] = [(a+b)+kn] = [a+b] = [a]\oplus[b]$

and similarly for any representative of $[b]$.

## Well-defined notation

For real numbers, the product $a \times b \times c$ is unambiguous because $(ab)c= a(bc)$.[1] In this case this notation is said to be well-defined. However, if the operation (here $\times$) did not have this property, which is known as associativity, then there must be a convention for which two elements to multiply first. Otherwise, the product is not well-defined. The subtraction operation, $-$, is not associative, for instance. However, the notation $a-b-c$ is well-defined under the convention that the $-$ operation is understood as addition of the opposite, thus $a-b-c$ is the same as $a + -b + -c$. Division is also non-associative. However, $a/b/c$ does not have an unambiguous conventional interpretation, so this expression is ill-defined.

## Other uses of the term

A solution to a partial differential equation is said to be well-defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed.[1]