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. 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.

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.

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

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

For real numbers, the product $a\times b\times c$ is unambiguous because $(ab)c=a(bc)$. 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

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

References

Books

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