Well-defined

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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.

[a]\oplus[b] = [a+b]

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

[a+kn]\oplus[b] = [(a+kn)+b] = [(a+b)+kn] = [a+b] = [a]\oplus[b]

and similarly for any representative of [b].

Well-defined notation[edit]

For real numbers, the product a \times b \times c is unambiguous because (ab)c= a(bc). [1] In this case this notation is said to be well-defined. However, if the operation (here \times) 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 a-b-c is well-defined under the convention that the - operation is understood as addition of the opposite, thus a-b-c is the same as a + -b + -c. Division is also non-associative. However, a/b/c does not have an unambiguous conventional interpretation, so this expression is ill-defined.

See also[edit]

References[edit]

Notes[edit]

  1. ^ a b 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.

Books[edit]

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