# Well-defined

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In mathematics, an expression is **well-defined** if it is unambiguous and its objects are independent of their representation. 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) is changed but the value of an input is not changed. A well-defined function gives the same output for 0.5 that it gives for 1/2.^{[1]} 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.^{[2]}

## Well-defined functions

All functions are well-defined binary relations: if there exist two ordered pairs in the function with the same first coordinate, then the two second coordinates must be equal. More precisely, if *(x,y)* and *(x,z)* are elements the function *f*, then *y=z*. Because the output assigned to *x* is unique in this sense, it is acceptable to use the notation *f(x)=y* (and/or *f(x)=z*) and to take advantage of the symmetric and transitive properties of equality. Thus if *f(x)=y* and *f(x)=z*, then of course *y=z*.

An equivalent way of expressing the definition above is this: given two ordered pairs *(a,b)* and *(c,d)*, the function *f* is *well-defined* iff whenever *a=c* it is the case that *b=d*. The contrapositive of this statement, which is equivalent and sometimes easier to use, says that *b≠d* implies *a≠c*. In other words, "different outputs must come from different inputs."

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 . ^{[2]} 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

- Equivalence relation#Well-definedness under an equivalence relation
- Definitionism
- Existence
- Uniqueness
- Uniqueness quantification
- Undefined

## References

### Notes

- ↑ 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. - ↑
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### Books

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