Wikipedia / Well-defined expression
Wikipedia / Well-defined expression
In mathematics, 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. 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 is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain.
Example
.
would be well defined and equal to \operatorname{mod}(a,2).
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
could be broken down into two logical steps: is a function if and only if – as a function – is well defined.
and not functional (as defined in (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 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 representativess. This is sometimes unavoidable when the arguments are cosets 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:
are the integers modulo m denotes the congruence class of n mod m.
is the argument of .
The function is well defined, because:
As a counter example, the converse definition:
.
Operations
In particular, the term well-defined is used with respect to (binary) operationss 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.
is an integer. Therefore,
the same, irrespective of the choice of representative.
Well-defined notation
; hence the notation is said to be well defined. 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, thus it is considered "well-defined". On the other hand, division, 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 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 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.
See also
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.