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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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''. 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). 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 ''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 Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
of ''f''.


Example

Let A_0,A_1 be sets, let A = A_0 \cup A_1 and "define" f: A \rightarrow \ as f(a)=0 if a \in A_0 and f(a)=1 if a \in A_1. Then f is well defined if A_0 \cap A_1 = \emptyset\!. For example, if A_0:=\ and A_1:=\, then f(a) would be well defined and equal to \operatorname(a,2). However, if A_0 \cap A_1 \neq \emptyset, then f would not be well defined because f(a) is "ambiguous" for a \in A_0 \cap A_1. For example, if A_0:=\ and A_1:=\, then f(2) would have to be both 0 and 1, which makes it ambiguous. As a result, the latter ''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 f could be broken down into two simple 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, f is a function if and only if A_0 \cap A_1 = \emptyset, in which case f – as a function – is well defined. On the other hand, if A_0 \cap A_1 \neq \emptyset, then for an a \in A_0 \cap A_1, we would have that (a,0) \in f ''and'' (a,1) \in f, which makes the binary relation f not ''functional'' (as defined in Binary relation#Special types of binary relations) and thus not well defined as a function. Colloquially, the "function" f is also called ambiguous at point 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 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
representative Representative may refer to: Politics *Representative democracy, type of democracy in which elected officials represent a group of people *House of Representatives, legislative body in various countries or sub-national entities *Legislator, someone ...
s. This is sometimes unavoidable when the arguments are
coset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and 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 : \begin f : & \Z/8\Z & \to & \Z/4\Z\\ & \overline_8 & \mapsto & \overline_4, \end where n\in\Z, m\in \ and \Z/m\Z are the integers modulo ''m'' and \overline_m denotes the
congruence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of ''n'' mod ''m''. N.B.: \overline_4 is a reference to the element n \in \overline_8, and \overline_8 is the argument of ''f''. The function ''f'' is well defined, because :n \equiv n' \bmod 8 \; \Leftrightarrow \; 8 \text (n-n') \Rightarrow \; 4 \text (n-n') \; \Leftrightarrow \; n \equiv n' \bmod 4. As a counter example, the converse definition : \begin g : & \Z/4\Z & \to & \Z/8\Z\\ & \overline_4 & \mapsto & \overline_8, \end does not lead to a well defined function, since e.g. \overline_4 equals \overline_4 in \Z/4\Z, but the first would be mapped by g to \overline_8, while the second would be mapped to \overline_8, and \overline_8 and \overline_8 are unequal in \Z/8\Z.


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. : oplus = +b/math> The fact that this is well defined follows from the fact that we can write any representative of /math> as a+kn, where k is an integer. Therefore, : oplus = +knoplus = a+kn)+b= a+b)+kn= +b and similarly for any representative of /math>, thereby making +b/math> the same irrespective of the choice of representative.


Well-defined notation

For real numbers, the product a \times b \times c is unambiguous because (a \times b)\times c = a \times (b \times c) (and hence the notation is said to be ''well defined''). This property, also known as
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
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 is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
operation, on the other hand, is not associative. However, there is a convention that a-b-c is shorthand for (a-b)-c, thus it is "well defined".
Division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
is also non-associative. However, in the case of a/b/c, parenthesization conventions are not so well established, so this expression is often considered ill defined. 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 the operator - for subtraction is ''left-to-right-associative'', which means that a-b-c is defined as (a-b)-c, and the operator = for assignment is ''right-to-left-associative'', which means that a=b=c is defined as a=(b=c). In the programming language APL there is only one rule: from – but parentheses first.


Other uses of the term

A solution to a
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is said to be well defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed.


See also

* *
DefinitionismDefinitionism (also called the classical theory of concepts) is the school of thought in which it is believed that a proper explanation of a theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the res ...
*
Existence Existence is the ability of an entity to interact with physical reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary Imaginary may refer to: * Imaginary (sociolog ...

Existence
*
Uniqueness Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison. When used in relation to humans, it is often in relation to a person's personality, or some specific characteristics of it, signalling that it i ...

Uniqueness
*
Uniqueness quantification In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
* Undefined


References


Notes


Sources

* ''Contemporary Abstract Algebra'', Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, . * ''Algebra: Chapter 0'', Paolo Aluffi, . Page 16. * ''Abstract Algebra'', Dummit and Foote, 3rd edition, . Page 1. {{DEFAULTSORT:well defined Definition Mathematical terminology