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Well-defined
In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous.[1] A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus: not a function).[2] The term well-defined is also used to indicate whether a logical statement is unambiguous.
Contents
Well-defined functions
Let \( A_1, A_2 \) be sets, let \( A = A_1 \bigcup A_2\) and define \( f: A \rightarrow \{1,2\} \) as f(a)=1 if \(a \in A_1\) and f(a)=2 if \(a \in A_2 \) . Then f is well-defined if \(A_1 \bigcap A_2 = \emptyset \) . If however \(A_1 \bigcap A_2 \neq \emptyset \) then f is not well-defined because f(a) is ambiguous for \(a \in A_1 \bigcap A_2\) .
In group theory, the term well-defined is often used when dealing with cosets, where a function f on a quotient group may be defined in terms of a coset representative. Here, a necessary requirement for f to be considered a function is that the output must be independent of which coset representative is chosen. The phrase f is well-defined is used to indicate that this requirement has been verified.
For example, consider \(\mathbb{Z}/2\mathbb{Z}\) , the integers modulo 2. Since 4 and 6 are congruent modulo 2, a function f whose domain is \(\mathbb{Z}/2\mathbb{Z} \) must give the same output when the input is represented by 4 that it gives when the input is represented by 6.
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).[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.
Other uses of the term
A solution to a partial differential equation is said to be well-defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed.[1]
See also
Equivalence relation ยง Well-definedness under an equivalence relation
Definitionism
Existence
Uniqueness
Uniqueness quantification
Undefined
References
Notes
Weisstein, Eric W. "Well-Defined". From MathWorld--A Wolfram Web Resource. Retrieved 2 January 2013.
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.
Algebra: Chapter 0, Paolo Aluffi, ISBN 978-0821847817. Page 16.
Abstract Algebra, Dummit and Foote, 3rd edition, ISBN 978-0471433347. Page 1.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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