Editing Well-defined expression (section)

{{Short description|Expression whose definition assigns it a unique interpretation}}
{{Other uses|Definition (disambiguation)}}

In [[mathematics]], a '''well-defined expression''' or '''unambiguous expression''' is an [[expression (mathematics)|expression]] whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', '''ill defined<!--boldface per WP:R#PLA-->''' or ''ambiguous''.<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 | access-date = 2 January 2013 }}</ref> 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 <math>f</math> takes real numbers as input, and if <math>f(0.5)</math> does not equal <math>f(1/2)</math> then <math>f</math> is not well defined (and thus not a function).<ref>Joseph J. Rotman, ''The Theory of Groups: an Introduction'', p.&nbsp;287 "... a function is "single-valued," or, as we prefer to say ... a function is ''well defined''.", Allyn and Bacon, 1965.</ref> The term ''well-defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory.

A function that is not well defined is not the same as a function that is [[undefined (mathematics)|undefined]]. For example, if <math>f(x)=\frac{1}{x}</math>, then even though <math>f(0)</math> is undefined, this does not mean that the function is ''not'' well defined; rather, 0 is not in the [[Domain of a function|domain]] of <math>f</math>.