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In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined, ill-defined or ambiguous.[1] 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).[2] The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.[3]

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 of f.

ExampleEdit

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 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 definitionEdit

In order to avoid the apostrophes around "define" in the previous simple example, the "definition" of   could be broken down into two simple logical steps:

  1. The definition of the binary relation: In the example
     ,
    (which so far is nothing but a certain subset of the Cartesian product  .)
  2. The assertion: The binary relation   is a function; in the example
     .

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:

  1. It provides a handy shorthand of the two-step approach.
  2. The relevant mathematical reasoning (i.e., step 2) is the same in both cases.
  3. In mathematical texts, the assertion is "up to 100%" true.

Independence of representativeEdit

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. This is sometimes unavoidable when the arguments are cosets and the equation refers to coset representatives.

Functions with one argumentEdit

For example, consider the following function

 

where   and   are the integers modulo m and   denotes the congruence class of n mod m.

N.B.:   is a reference to the element  , and   is the argument of  .

The function   is well-defined, because

 

OperationsEdit

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.

 

The fact that this is well-defined follows from the fact that we can write any representative of   as  , where   is an integer. Therefore,

 

and similarly for any representative of  , thereby making   the same irrespective of the choice of representative.[3]

Well-defined notationEdit

For real numbers, the product   is unambiguous because   (and hence the notation is said to be well-defined).[1] This property, also known as associativity 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 operation, on the other hand, is not associative. However, there is a convention (or definition) in that the   operation is understood as addition of additive inverse, thus   is the same as  , and is thus "well-defined".

Division is also non-associative. However, in the case of   the convention   is not so well established, so this expression is 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).[4] In the programming language APL there is only one rule: from right to left — but parentheses first.

Other uses of the termEdit

A solution to a partial differential equation is said to be well-defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed.[1]

See alsoEdit

ReferencesEdit

NotesEdit

  1. ^ a b c Weisstein, Eric W. "Well-Defined". From MathWorld--A Wolfram Web Resource. Retrieved 2 January 2013.
  2. ^ Joseph J. Rotman, The Theory of Groups: an Introduction, p. 287 "... a function is "single-valued," or, as we prefer to say ... a function is well defined.", Allyn and Bacon, 1965.
  3. ^ a b "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2019-10-18.
  4. ^ "Operator Precedence and Associativity in C". GeeksforGeeks. 2014-02-07. Retrieved 2019-10-18.

SourcesEdit