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In mathematics, an expression is welldefined 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 welldefined 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 welldefined is also used to indicate whether a logical statement is unambiguous, and a solution to a partial differential equation is said to be welldefined if it is continuous on the boundary.^{[1]}
Welldefined functions
In mathematics, a function is welldefined 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 welldefined 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 welldefined 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 welldefined 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 welldefined. It is; 0 is simply not in the domain of the function.
Operations
In particular, the term welldefined 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 welldefined 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 welldefined follows from the fact that we can write any representative of as , where k is an integer. Therefore,
and similarly for any representative of .
Welldefined notation
For real numbers, the product is unambiguous because . ^{[1]} In this case this notation is said to be welldefined. 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 welldefined. The subtraction operation, , is not associative, for instance. However, the notation is welldefined under the convention that the operation is understood as addition of the opposite, thus is the same as . Division is also nonassociative. However, does not have an unambiguous conventional interpretation, so this expression is illdefined.
See also
References
Notes
 ↑ ^{1.0} ^{1.1} Template:Cite web
 ↑ Joseph J. Rotman, The Theory of Groups: an Introduction, p.287 "...a function is "singlevalued," 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 0618514716.