In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill 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 ${\displaystyle f}$ takes real numbers as input, and if ${\displaystyle f(0.5)}$ does not equal ${\displaystyle f(1/2)}$ then ${\displaystyle f}$ is not well defined (and thus not a function).^[2] The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.

A function that is not well defined is not the same as a function that is undefined. For example, if ${\displaystyle f(x)={\frac {1}{x}}}$, then even though ${\displaystyle f(0)}$ is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of ${\displaystyle f}$.

Example

Let ${\displaystyle A_{0},A_{1}}$ be sets, let ${\displaystyle A=A_{0}\cup A_{1}}$ and "define" ${\displaystyle f:A\rightarrow \{0,1\}}$ as ${\displaystyle f(a)=0}$ if ${\displaystyle a\in A_{0}}$ and ${\displaystyle f(a)=1}$ if ${\displaystyle a\in A_{1}}$.

Then ${\displaystyle f}$ is well defined if ${\displaystyle A_{0}\cap A_{1}=\emptyset \!}$. For example, if ${\displaystyle A_{0}:=\{2,4\}}$ and ${\displaystyle A_{1}:=\{3,5\}}$, then ${\displaystyle f(a)}$ would be well defined and equal to ${\displaystyle \operatorname {mod} (a,2)}$.

However, if ${\displaystyle A_{0}\cap A_{1}\neq \emptyset }$, then ${\displaystyle f}$ would not be well defined because ${\displaystyle f(a)}$ is "ambiguous" for ${\displaystyle a\in A_{0}\cap A_{1}}$. For example, if ${\displaystyle A_{0}:=\{2\}}$ and ${\displaystyle A_{1}:=\{2\}}$, then ${\displaystyle f(2)}$ would have to be both 0 and 1, which makes it ambiguous. As a result, the latter ${\displaystyle f}$ is not well defined and thus not a function.

"Definition" as anticipation of definition

In order to avoid the quotation marks around "define" in the previous simple example, the "definition" of ${\displaystyle f}$ could be broken down into two logical steps:

While the definition in step 1 is formulated with the freedom of any definition and is certainly effective (without the need to classify it as "well defined"), the assertion in step 2 has to be proved. That is, ${\displaystyle f}$ is a function if and only if ${\displaystyle A_{0}\cap A_{1}=\emptyset }$, in which case ${\displaystyle f}$ – as a function – is well defined. On the other hand, if ${\displaystyle A_{0}\cap A_{1}\neq \emptyset }$, then for an ${\displaystyle a\in A_{0}\cap A_{1}}$, we would have that ${\displaystyle (a,0)\in f}$ and ${\displaystyle (a,1)\in f}$, which makes the binary relation ${\displaystyle f}$ not functional (as defined in Binary relation#Special types of binary relations) and thus not well defined as a function. Colloquially, the "function" ${\displaystyle f}$ is also called ambiguous at point ${\displaystyle a}$ (although there is per definitionem never an "ambiguous function"), and the original "definition" is pointless.

Despite these subtle logical problems, it is quite common to use the term definition (without apostrophes) for "definitions" of this kind, for three reasons:

1. It provides a handy shorthand of the two-step approach.
2. The relevant mathematical reasoning (i.e., step 2) is the same in both cases.
3. In mathematical texts, the assertion is "up to 100%" true.

Independence of representative

Questions regarding the well-definedness of a function often arise when the defining equation of a function refers not only to the arguments themselves, but also to elements of the arguments, serving as representatives. This is sometimes unavoidable when the arguments are cosets and when the equation refers to coset representatives. The result of a function application must then not depend on the choice of representative.

Functions with one argument

For example, consider the following function:

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