Well-defined
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 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 is also used to indicate whether a logical statement is unambiguous.
Well-defined functions
Let be sets, let
and define
as
if
and
if
. Then
is well-defined if
. If however
then
is not well-defined because
is ambiguous for
.
In group theory, the term well-defined is often used when dealing with cosets, where a function f on a quotient group may be defined in terms of a coset representative. Here, a necessary requirement for f to be considered a function is that the output must be independent of which coset representative is chosen. The phrase f is well-defined is used to indicate that this requirement has been verified.
For example, consider , the integers modulo 2. Since 4 and 6 are congruent modulo 2, a function f whose domain is
must give the same output when the input is represented by 4 that it gives when the input is represented by 6.
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 f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined. It is; 0 is simply not in the domain of the function.
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.
The fact that this is well-defined follows from the fact that we can write any representative of as
, where k is an integer. Therefore,
and similarly for any representative of .
Well-defined notation
For real numbers, the product is unambiguous because
.[1] In this case this notation is said to be well-defined. However, if the operation (here
) did not have this property, which is known as associativity, then there must be a convention for which two elements to multiply first. Otherwise, the product is not well-defined. The subtraction operation,
, is not associative, for instance. However, the notation
is well-defined under the convention that the
operation is understood as addition of the opposite, thus
is the same as
. Division is also non-associative. However,
does not have an unambiguous conventional interpretation, so this expression is ill-defined.
Other uses of the term
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 also
- Equivalence relation § Well-definedness under an equivalence relation
- Definitionism
- Existence
- Uniqueness
- Uniqueness quantification
- Undefined
References
Notes
- ↑ 1.0 1.1 1.2 Weisstein, Eric W. "Well-Defined". From MathWorld--A Wolfram Web Resource. Retrieved 2 January 2013.
- ↑ 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.
Books
- Contemporary Abstract Algebra, Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, ISBN 0-618-51471-6.
- Algebra: Chapter 0, Paolo Aluffi, ISBN 978-0821847817. Page 16.
- Abstract Algebra, Dummit and Foote, 3rd edition, ISBN 978-0471433347. Page 1.