Editing Well-defined expression

{{Short description|Expression whose definition assigns it a unique interpretation}}
{{Other uses|Definition (disambiguation)}}

In [[mathematics]], a '''well-defined expression''' or '''unambiguous expression''' is an [[expression (mathematics)|expression]] whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', '''ill defined<!--boldface per WP:R#PLA-->''' 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 <math>f</math> takes real numbers as input, and if <math>f(0.5)</math> does not equal <math>f(1/2)</math> then <math>f</math> 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.

A function that is not well defined is not the same as a function that is [[undefined (mathematics)|undefined]]. For example, if <math>f(x)=\frac{1}{x}</math>, then even though <math>f(0)</math> is undefined, this does not mean that the function is ''not'' well defined; rather, 0 is not in the [[Domain of a function|domain]] of <math>f</math>.

==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 quotation marks around "define" in the previous simple example, the "definition" of <math>f</math> could be broken down into two logical steps:
{{ordered list
| ''The definition'' of the [[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 proven. 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#Special types of binary relations]]) 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. <br> <br />

Despite these subtle logical problems, it is quite common to 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==
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 [[representative (mathematics)|representative]]s. This is sometimes unavoidable when the arguments are [[coset]]s 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:
:<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>

similar holds for any representative of <math>[b]</math>, thereby making <math>[a+b]</math> the same, irrespective of the choice of representative.

==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>; hence the notation is said to be ''well defined''.<ref name="MathWorld"/> This property, also known as [[associativity]] of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore, a specification of the sequence can be omitted. The [[subtraction]] operation is non-associative; despite that, there is a convention that <math>a-b-c</math> is shorthand for <math>(a-b)-c</math>, thus it is considered "well-defined". On the other hand, [[Division (mathematics)|Division]] is non-associative, and in the case of <math>a/b/c</math>, parenthesization conventions are not well established; therefore, this expression is often considered ill-defined.

Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of [[Operator precedence|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]] is said to be ''well-defined'' if it is continuously determined by boundary conditions as those boundary conditions are changed.<ref name="MathWorld" />

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

==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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[[Category:Definition]]
[[Category:Mathematical terminology]]