In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a well-defined expression or unambiguous expression is an
expression
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, o ...
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
Undefined may refer to:
Mathematics
* Undefined (mathematics), with several related meanings
** Indeterminate form, in calculus
Computing
* Undefined behavior, computer code whose behavior is not specified under certain conditions
* 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, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
of ''f''.
Example
Let
be sets, let
and "define"
as
if
and
if
.
Then
is well defined if
. For example, if
and
, then
would be well defined and equal to
.
However, if
, then
would not be well defined because
is "ambiguous" for
. For example, if
and
, then
would have to be both 0 and 1, which makes it ambiguous. As a result, the latter ''
'' 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
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,
is a function if and only if
, in which case
– as a function – is well defined.
On the other hand, if
, then for an
, we would have that
''and''
, which makes the binary relation
not ''functional'' (as defined in
Binary relation#Special types of binary relations) and thus not well defined as a function. Colloquially, the "function"
is also called ambiguous at point
(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, som ...
s. This is sometimes unavoidable when the arguments are
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
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
:
where
and
are the
integers modulo ''m'' and
denotes the
congruence class
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
of ''n'' mod ''m''.
N.B.:
is a reference to the element
, and
is the argument of ''
''.
The function ''
'' is well defined, because
:
As a counter example, the converse definition
:
does not lead to a well defined function, since e.g.
equals
in
, but the first would be mapped by
to
, while the second would be mapped to
, and
and
are unequal in
.
Operations
In particular, the term ''well defined'' is used with respect to (binary)
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
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.
: