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| {{Other uses|Definition (disambiguation)}}
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| 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 [[continuity (mathematics)|continuous]] on the boundary.<ref name="MathWorld" />
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| ==Well-defined functions==
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| 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 number]]s 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.<ref> 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.</ref>
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| In [[group theory]], the term well-defined is often used when dealing with [[coset]]s, 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 [[modulo arithmetic|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.
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| A function that is not well-defined is not the same as a function that is [[undefined (mathematics)|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.
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| ===Operations===
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| In particular, the term well-defined is used with respect to (binary) [[operation (mathematics)| operation]]s 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.
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| :<math>[a]\oplus[b] = [a+b]</math> | |
| The fact that this is well-defined follows from the fact that we can write any representative of <math>[a]</math> as <math>a+kn</math>, where k is an integer. Therefore,
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| :<math>[a+kn]\oplus[b] = [(a+kn)+b] = [(a+b)+kn] = [a+b] = [a]\oplus[b]</math>
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| and similarly for any representative of <math>[b]</math>.
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| ==Well-defined notation==
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| For real numbers, the product <math>a \times b \times c</math> is unambiguous because <math>(ab)c= a(bc)</math>. <ref name="MathWorld">{{cite web | last = Weisstein | first = Eric W. | title = Well-Defined | publisher = From MathWorld--A Wolfram Web Resource | url=http://mathworld.wolfram.com/Well-Defined.html | accessdate = 2 January 2013 }}</ref> In this case this notation is said to be ''well-defined''. However, if the [[operation (mathematics)|operation]] (here <math>\times</math>) 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, <math>-</math>, is not associative, for instance. However, the notation <math>a-b-c</math> is well-defined under the convention that the <math>-</math> operation is understood as addition of the opposite, thus <math>a-b-c</math> is the same as <math>a + -b + -c</math>. Division is also non-associative. However, <math>a/b/c</math> does not have an unambiguous conventional interpretation, so this expression is ill-defined.
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| ==See also==
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| * [[Definitionism]]
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| * [[Existence]]
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| * [[Uniqueness]]
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| * [[Uniqueness quantification]]
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| * [[Undefined_(mathematics)|Undefined]]
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| ==References==
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| ===Notes===
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| {{Reflist}}
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| ===Books===
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| * ''Contemporary Abstract Algebra'', Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, ISBN 0-618-51471-6.
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| {{DEFAULTSORT:Well-Defined}}
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| [[Category:Definition]]
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| [[Category:Mathematical terminology]]
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