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 may refer to:
*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 ...
be sets, let
is well defined if
. For example, if
would be well defined and equal to
would not be well defined because
is "ambiguous" for
. For example, if
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
, 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 may refer to:
*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
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
are the integers modulo ''m''
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''.
is a reference to the element
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.
, but the first would be mapped by
, while the second would be mapped to
are unequal in
In particular, the term ''well defined'' is used with respect to (binary) 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.