Difference between revisions of "Range"
m |
|||
Line 1: | Line 1: | ||
Let <math>A</math> and <math>B</math> be any [[set]]s and let <math>f:A\to B</math> be any [[function]] between them, so that <math>A</math> is the [[domain]] of <math>f</math> and <math>B</math> is the [[codomain]]. Then <math>\{b\in B\mid \mathrm{there\ is\ some\ } a\in A\mathrm{\ such\ that\ } f(a)=b\}</math> is called the '''range''' or '''image''' of <math>f</math>. | Let <math>A</math> and <math>B</math> be any [[set]]s and let <math>f:A\to B</math> be any [[function]] between them, so that <math>A</math> is the [[domain]] of <math>f</math> and <math>B</math> is the [[codomain]]. Then <math>\{b\in B\mid \mathrm{there\ is\ some\ } a\in A\mathrm{\ such\ that\ } f(a)=b\}</math> is called the '''range''' or '''image''' of <math>f</math>. | ||
− | Thus, if we have <math>f: \mathbb{R} \to \mathbb{R}</math> given by <math>f(x) = x^2</math>, the range of <math>f</math> is the set of [[nonnegative]] | + | Thus, if we have <math>f: \mathbb{R} \to \mathbb{R}</math> given by <math>f(x) = x^2</math>, the range of <math>f</math> is the set of [[nonnegative]] [[real number]]s. |
A function is a [[surjection]] exactly when the range is equal to the codomain. | A function is a [[surjection]] exactly when the range is equal to the codomain. |
Revision as of 11:30, 31 July 2006
Let and
be any sets and let
be any function between them, so that
is the domain of
and
is the codomain. Then
is called the range or image of
.
Thus, if we have given by
, the range of
is the set of nonnegative real numbers.
A function is a surjection exactly when the range is equal to the codomain.