Well-defined

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In mathematics, an expression is well-defined if it is unambiguous and its objects are independent of their representative. More simply, it means that a mathematical statement is sensible and definite. In particular, a function is well-defined if it gives the same result when the form (the way in which it is presented) but not the value of an input is changed. The term well-defined is also used to indicate whether a logical statement is unambiguous, and a solution to a partial differential equation is said to be well-defined if it is continuous on the boundary.[1]

Well-defined functions

In mathematics, a function is well-defined if it gives the same result when the form but not the value of the input is changed. For example, a function on the real numbers must give the same output for 0.5 as it does for 1/2, because in the real number system 0.5 = 1/2. An example of a "function" that is not well-defined is "f(x) = the first digit that appears in x". For this function, f(0.5) = 0 but f(1/2) = 1. A "function" such as this would not be considered a function at all, since a function must have exactly one output for a given input.[2]

In group theory, the term well-defined is often used when dealing with cosets, where a function on a quotient group may be defined in terms of a coset representative. Then the output of the function must be independent of which coset representative is chosen. For example, consider the group of integers modulo 2. Since 4 and 6 are congruent modulo 2, a function defined on the integers modulo 2 must give the same output when the input is 6 that it gives when the input is 4.

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.

See also

References

Notes

  1. 1.0 1.1 Template:Cite web
  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.

Books

  • Contemporary Abstract Algebra, Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, ISBN 0-618-51471-6.