Editing Image (mathematics) (section)

==Definition==
{{Group theory sidebar |Basics}}
[[File:ImagePreimageOfElement.png|300px|thumb|right|<math>f</math> is a function from domain <math>X</math> to codomain <math>Y</math>. The image of element <math>x</math> is element <math>y</math>. The preimage of element <math>y</math> is the set {<math>x, x'</math>}. The preimage of element <math>y'</math> is <math>\varnothing</math>.]]
[[File:ImagePreimageofaSet.png|300px|thumb|right|<math>f</math> is a function from domain <math>X</math> to codomain <math>Y</math>. The image of all elements in subset <math>A</math> is subset <math>B</math>. The preimage of <math>B</math> is subset <math>C</math>]]
[[File:Codomain2.SVG|thumb|upright=1.2|<math>f</math> is a function from domain <math>X</math> to codomain <math>Y.</math> The yellow oval inside <math>Y</math> is the image of <math>f</math>. The preimage of <math>Y</math> is the entire domain <math>X</math>]]
The word "image" is used in three related ways. In these definitions, <math>f : X \to Y</math> is a [[Function (mathematics)|function]] from the [[Set (mathematics)|set]] <math>X</math> to the set <math>Y.</math>

===Image of an element===

If <math>x</math> is a member of <math>X,</math> then the image of <math>x</math> under <math>f,</math> denoted <math>f(x),</math> is the [[Value (mathematics)|value]] of <math>f</math> when applied to <math>x.</math> <math>f(x)</math> is alternatively known as the output of <math>f</math> for argument <math>x.</math> 

Given <math>y,</math> the function <math>f</math> is said to {{em|take the value <math>y</math>}} or {{em|take <math>y</math> as a value}} if there exists some <math>x</math> in the function's domain such that <math>f(x) = y.</math> 
Similarly, given a set <math>S,</math> <math>f</math> is said to {{em|take a value in <math>S</math>}} if there exists {{em|some}} <math>x</math> in the function's domain such that <math>f(x) \in S.</math> 
However, {{em|<math>f</math> takes [all] values in <math>S</math>}} and {{em|<math>f</math> is valued in <math>S</math>}} means that <math>f(x) \in S</math> for {{em|every}} point <math>x</math> in the domain of <math>f</math> .

===Image of a subset===

Throughout, let <math>f : X \to Y</math> be a function. 
The {{anchor|image of a set}}{{em|image}} under <math>f</math> of a subset <math>A</math> of <math>X</math> is the set of all <math>f(a)</math> for <math>a\in A.</math> It is denoted by <math>f[A],</math> or by <math>f(A)</math> when there is no risk of confusion. Using [[set-builder notation]], this definition can be written as<ref>{{Cite web|date=2019-11-05| title=5.4: Onto Functions and Images/Preimages of Sets| url=https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/5%3A_Functions/5.4%3A_Onto_Functions_and_Images%2F%2FPreimages_of_Sets| access-date=2020-08-28| website=Mathematics LibreTexts| language=en}}</ref><ref>{{cite book| author=Paul R. Halmos| title=Naive Set Theory| location=Princeton| publisher=Nostrand| year=1968 }} Here: Sect.8</ref>
<math display=block>f[A] = \{f(a) : a \in A\}.</math>

This induces a function <math>f[\,\cdot\,] : \mathcal P(X) \to \mathcal P(Y),</math> where <math>\mathcal P(S)</math> denotes the [[power set]] of a set <math>S;</math> that is the set of all [[subset]]s of <math>S.</math> See {{Section link||Notation}} below for more.

===Image of a function===

The ''image'' of a function is the image of its entire [[Domain of a function|domain]], also known as the [[Range of a function|range]] of the function.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Image|url=https://mathworld.wolfram.com/Image.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> This last usage should be avoided because the word "range" is also commonly used to mean the [[codomain]] of <math>f.</math>

===Generalization to binary relations===

If <math>R</math> is an arbitrary [[binary relation]] on <math>X \times Y,</math> then the set <math>\{ y \in Y : x R y \text{ for some } x \in X \}</math> is called the image, or the range, of <math>R.</math> Dually, the set <math>\{ x \in X : x R y \text{ for some } y \in Y \}</math> is called the domain of <math>R.</math>