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In [[Mathematics|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'', ''ill-defined'' or ''ambiguous''.<ref name="MathWorld">{{cite web | last = Weisstein | first = Eric W. | title = Well-Defined | publisher = From MathWorld--A Wolfram Web Resource | url=http://mathworld.wolfram.com/Well-Defined.html | access-date = 2 January 2013 }}</ref> 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).<ref>Joseph J. Rotman, ''The Theory of Groups: an Introduction'', p.&nbsp;287 "... a function is "single-valued," or, as we prefer to say ... a function is ''well defined''.", Allyn and Bacon, 1965.</ref> The term ''well-defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory.<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/|title=The Definitive Glossary of Higher Mathematical Jargon|date=2019-08-01|website=Math Vault|language=en-US|access-date=2019-10-18}}</ref>

A function that is not well-defined is not the same as a function that is [[Undefined (mathematics)|undefined]]. For example, if ''f''(''x'') = 1/''x'', then the fact that ''f''(0) is undefined does not mean that the ''f'' is ''not'' well-defined —  but that 0 is simply not in the [[Domain of a function|domain]] of ''f''.

==Example==
Let <math>A_0,A_1</math> be sets, let <math>A = A_0 \cup A_1</math> and "define" <math>f: A \rightarrow \{0,1\}</math> as <math>f(a)=0</math> if <math>a \in A_0</math> and <math>f(a)=1</math> if <math>a \in A_1</math>.

Then <math>f</math> is well-defined if <math>A_0 \cap A_1 = \emptyset\!</math>. For example, if <math>A_0:=\{2,4\}</math>and <math>A_1:=\{3,5\}</math>, then <math>f(a)</math>would be well-defined and equal to [[Modulo operation|<math>\operatorname{mod}(a,2)</math>]].

However, if <math>A_0 \cap A_1 \neq \emptyset</math>, then <math>f</math> would not be well-defined because <math>f(a)</math> is "ambiguous" for <math>a \in A_0 \cap A_1</math>. For example, if <math>A_0:=\{2\}</math> and <math>A_1:=\{2\}</math>, then <math>f(2)</math> would have to be both 0 and 1, which makes it ambiguous. As a result, the latter ''<math>f</math>'' is not well-defined and thus not a function.

=="Definition" as anticipation of definition==
In order to avoid the apostrophes around "define" in the previous simple example, the "definition" of <math>f</math> could be broken down into two simple logical steps:
{{ordered list
| ''The definition'' of the [[Binary relation|binary relation]]: In the example
:<math>f := \bigl\{(a,i) \mid i \in \{0,1\} \wedge a \in A_i \bigr\} </math>,
(which so far is nothing but a certain subset of the [[Cartesian product]] <math>A \times \{0,1\}</math>.)
| ''The assertion'': The binary relation <math>f</math> is a function; in the example
:<math>f: A \rightarrow \{0,1\}</math>.
}}
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, <math>f</math> is a function if and only if <math>A_0 \cap A_1 = \emptyset</math>, in which case <math>f</math>— as a function — is well-defined.

On the other hand, if <math>A_0 \cap A_1 \neq \emptyset</math>, then for an <math>a \in A_0 \cap A_1</math>, we would have that <math>(a,0) \in f</math> ''and'' <math>(a,1) \in f</math>, which makes the binary relation <math>f</math> not ''functional'' (as defined in [[Binary relation]]) and thus not well-defined as a function. Colloquially, the "function" <math>f</math> is also called ambiguous at point <math>a</math> (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 anticipatorily use the term definition (without apostrophes) for "definitions" of this kind — for three reasons:

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

==Independence of representative==
The question of well-definedness of a function classically arises when the defining equation of a function does not (only) refer to the arguments themselves, but (also) to elements of the arguments, serving as representatives. This is sometimes unavoidable when the arguments are [[Coset|coset]]s and the equation refers to coset representatives. The result of a function application must then not depend on the choice of reprentative.

===Functions with one argument===
For example, consider the following function
:<math>
\begin{matrix}
f : & \Z/8\Z         & \to     & \Z/4\Z\\
    & \overline{n}_8 & \mapsto & \overline{n}_4,
\end{matrix}</math>
where <math>n\in\Z, m\in \{4,8\}</math> and <math>\Z/m\Z</math> are the [[Modular arithmetic|integers modulo ''m'']] and <math>\overline{n}_m</math> denotes the [[Modular arithmetic#Congruence classes|congruence class]] of ''n'' mod ''m''.

N.B.: <math>\overline{n}_4</math> is a reference to the element <math>n \in \overline{n}_8</math>, and <math>\overline{n}_8</math> is the argument of ''<math>f</math>''.

The function ''<math>f</math>'' is well-defined, because

:<math>n \equiv n' \bmod 8 \; \Leftrightarrow \; 8 \text{ divides } (n-n') \Rightarrow \; 4 \text{ divides } (n-n') \; \Leftrightarrow \; n \equiv n' \bmod 4.</math>

As a counter example, the converse definition

:<math>
\begin{matrix}
g : & \Z/4\Z         & \to     & \Z/8\Z\\
    & \overline{n}_4 & \mapsto & \overline{n}_8,
\end{matrix}</math>
does not lead to a  well-defined function, since e.g. <math>\overline{1}_4</math> equals <math>\overline{5}_4</math> in <math>\Z/4\Z</math>, but the first would be mapped by <math>g</math> to <math>\overline{1}_8</math>, while the second would be mapped to <math>\overline{5}_8</math>, and <math>\overline{1}_8</math> and <math>\overline{5}_8</math> are unequal in <math>\Z/8\Z</math>.

===Operations===
In particular, the term well-defined is used with respect to (binary) [[Operation (mathematics)|operation]]s 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. 
:<math>[a]\oplus[b] = [a+b]</math>

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

:<math>[a]\oplus[b] = [a+kn]\oplus[b] = [(a+kn)+b] = [(a+b)+kn] = [a+b];</math>

and similarly for any representative of <math>[b]</math>, thereby making <math>[a+b]</math> the same irrespective of the choice of representative.<ref name=":0" />

==Well-defined notation==
For real numbers, the product <math>a \times b \times c</math> is unambiguous because <math>(a \times b)\times c = a \times (b \times c)</math> (and hence the notation is said to be ''well-defined'').<ref name="MathWorld"/> This property, also known as associativity of multiplication, guarantees that the result does not depend on the sequence of multiplications, so that a specification of the sequence can be omitted.

The [[Subtraction|subtraction]] operation, on the other hand, is not associative. However, there is a convention that <math>a-b-c</math> is shorthand for <math>(a-b)-c</math>, thus it is "well-defined".

[[Division (mathematics)|Division]] is also non-associative. However, in the case of <math>a/b/c</math>, parenthezation conventions are not so well established, so this expression is often considered '''ill-defined'''<!--boldface per WP:R#PLA-->.

Unlike with functions, the notational ambiguities can be overcome more or less easily by means of additional definitions (e.g., rules of precedence, associativity of the operator). For example, in the programming language [[C (programming language)|C]] the operator <code>-</code> for subtraction is ''left-to-right-associative'', which means that <code>a-b-c</code> is defined as <code>(a-b)-c</code>, and the operator <code>=</code> for assignment is ''right-to-left-associative'', which means that <code>a=b=c</code> is defined as <code>a=(b=c)</code>.<ref>{{Cite web|url=https://www.geeksforgeeks.org/operator-precedence-and-associativity-in-c/|title=Operator Precedence and Associativity in C|date=2014-02-07|website=GeeksforGeeks|language=en-US|access-date=2019-10-18}}</ref> In the programming language [[APL (programming language)|APL]] there is only one rule: from [[APL (programming language)#Design|right to left]] — but parentheses first.

==Other uses of the term==
A solution to a [[Partial differential equation|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.<ref name="MathWorld" />

==See also==
* {{multi-section link|Equivalence relation|Well-definedness under an equivalence relation}}
* [[Definitionism]]
* [[Philosophy:Existence|Existence]]
* Uniqueness
* [[Uniqueness quantification]]
* [[Undefined (mathematics)|Undefined]]

==References==

===Notes===
{{Reflist}}

===Sources===
* ''Contemporary Abstract Algebra'', Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, {{isbn|0-618-51471-6}}.
* ''Algebra: Chapter 0'', Paolo Aluffi, {{isbn|978-0821847817}}. Page 16.
* ''Abstract Algebra'', Dummit and Foote, 3rd edition, {{isbn|978-0471433347}}. Page 1.

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