Image (mathematics)#Image of a subset

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In mathematics, for a function $f: X \to Y$ , the image of an input value $x$ is the single output value produced by $f$ when passed $x$ . The preimage of an output value $y$ is the set of input values that produce $y$ .

More generally, evaluating $f$ at each element of a given subset $A$ of its domain $X$ produces a set, called the "image of $A$ under (or through) $f$ ". Similarly, the inverse image (or preimage) of a given subset $B$ of the codomain $Y$ is the set of all elements of $X$ that map to a member of $B.$

The image of the function $f$ is the set of all output values it may produce, that is, the image of $X$ . The preimage of $f$ , that is, the preimage of $Y$ under $f$ , always equals $X$ (the domain of $f$ ); therefore, the former notion is rarely used.

Image and inverse image may also be defined for general binary relations, not just functions.

Definition

File:ImagePreimageOfElement.png

File:ImagePreimageofaSet.png

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The word "image" is used in three related ways. In these definitions, $f : X \to Y$ is a function from the set $X$ to the set $Y.$

=Image of an element=

If $x$ is a member of $X,$ then the image of $x$ under $f,$ denoted $f(x),$ is the value of $f$ when applied to $x.$ $f(x)$ is alternatively known as the output of $f$ for argument $x.$

Given $y,$ the function $f$ is said to {{em|take the value $y$ }} or {{em|take $y$ as a value}} if there exists some $x$ in the function's domain such that $f(x) = y.$

Similarly, given a set $S,$ $f$ is said to {{em|take a value in $S$ }} if there exists {{em|some}} $x$ in the function's domain such that $f(x) \in S.$

However, {{em| $f$ takes [all] values in $S$ }} and {{em| $f$ is valued in $S$ }} means that $f(x) \in S$ for {{em|every}} point $x$ in the domain of $f$ .

=Image of a subset=

Throughout, let $f : X \to Y$ be a function.

The {{anchor|image of a set}}{{em|image}} under $f$ of a subset $A$ of $X$ is the set of all $f(a)$ for $a\in A.$ It is denoted by $f[A],$ or by $f(A)$ when there is no risk of confusion. Using set-builder notation, this definition can be written as{{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}}{{cite book| author=Paul R. Halmos| title=Naive Set Theory| location=Princeton| publisher=Nostrand| year=1968 }} Here: Sect.8

$f[A] = \{f(a) : a \in A\}.$

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

=Image of a function=

The image of a function is the image of its entire domain, also known as the range of the function.{{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}} This last usage should be avoided because the word "range" is also commonly used to mean the codomain of $f.$

=Generalization to binary relations=

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

Inverse image

{{Redirect|Preimage|the cryptographic attack on hash functions|preimage attack}}

Let $f$ be a function from $X$ to $Y.$ The preimage or inverse image of a set $B \subseteq Y$ under $f,$ denoted by $f^{-1}[B],$ is the subset of $X$ defined by

$f^{-1}[ B ] = \{ x \in X \,:\, f(x) \in B \}.$

Other notations include $f^{-1}(B)$ and $f^{-}(B).$ {{sfn|Dolecki|Mynard|2016|pp=4-5}}

The inverse image of a singleton set, denoted by $f^{-1}[\{ y \}]$ or by $f^{-1}(y),$ is also called the fiber or fiber over $y$ or the level set of $y.$ The set of all the fibers over the elements of $Y$ is a family of sets indexed by $Y.$

For example, for the function $f(x) = x^2,$ the inverse image of $\{ 4 \}$ would be $\{ -2, 2 \}.$ Again, if there is no risk of confusion, $f^{-1}[B]$ can be denoted by $f^{-1}(B),$ and $f^{-1}$ can also be thought of as a function from the power set of $Y$ to the power set of $X.$ The notation $f^{-1}$ should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of $B$ under $f$ is the image of $B$ under $f^{-1}.$

<span id="Notation">Notation</span> for image and inverse image

The traditional notations used in the previous section do not distinguish the original function $f : X \to Y$ from the image-of-sets function $f : \mathcal{P}(X) \to \mathcal{P}(Y)$ ; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternative{{sfn|Blyth|2005|p=5}} is to give explicit names for the image and preimage as functions between power sets:

=Arrow notation=

$f^\rightarrow : \mathcal{P}(X) \to \mathcal{P}(Y)$ with $f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}$
$f^\leftarrow : \mathcal{P}(Y) \to \mathcal{P}(X)$ with $f^\leftarrow(B) = \{ a \in X \;|\; f(a) \in B\}$

=Star notation=

$f_\star : \mathcal{P}(X) \to \mathcal{P}(Y)$ instead of $f^\rightarrow$
$f^\star : \mathcal{P}(Y) \to \mathcal{P}(X)$ instead of $f^\leftarrow$

=Other terminology=

An alternative notation for $f[A]$ used in mathematical logic and set theory is $f\,''A.$ {{cite book| title=Set Theory for the Mathematician|url=https://archive.org/details/settheoryformath0000rubi|url-access=registration|author=Jean E. Rubin |author-link= Jean E. Rubin |page=xix|year=1967 |publisher=Holden-Day |asin=B0006BQH7S}}M. Randall Holmes: [https://web.archive.org/web/20180207010648/https://pdfs.semanticscholar.org/d8d8/5cdd3eb2fd9406d13b5c04d55708068031ef.pdf Inhomogeneity of the urelements in the usual models of NFU], December 29, 2005, on: Semantic Scholar, p. 2
Some texts refer to the image of $f$ as the range of $f,$ {{Cite book |last=Hoffman |first=Kenneth |title=Linear Algebra |publisher=Prentice-Hall |year=1971 |edition=2nd |pages=388 |language=en}} but this usage should be avoided because the word "range" is also commonly used to mean the codomain of $f.$

Examples

$f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}$ defined by

\left\{\begin{matrix} 1 \mapsto a, \\ 2 \mapsto a, \\ 3 \mapsto c. \end{matrix}\right. {{paragraph break}} The image of the set

\{ 2, 3 \}

under

f

f(\{ 2, 3 \}) = \{ a, c \}.

The image of the function

f

\{ a, c \}.

The preimage of

a

f^{-1}(\{ a \}) = \{ 1, 2 \}.

The preimage of

\{ a, b \}

is also

f^{-1}(\{ a, b \}) = \{ 1, 2 \}.

The preimage of

\{ b, d \}

under

f

is the empty set

\{ \ \} = \emptyset.

$f : \R \to \R$ defined by $f(x) = x^2.$ {{paragraph break}} The image of $\{ -2, 3 \}$ under $f$ is $f(\{ -2, 3 \}) = \{ 4, 9 \},$ and the image of $f$ is $\R^+$ (the set of all positive real numbers and zero). The preimage of $\{ 4, 9 \}$ under $f$ is $f^{-1}(\{ 4, 9 \}) = \{ -3, -2, 2, 3 \}.$ The preimage of set $N = \{ n \in \R : n < 0 \}$ under $f$ is the empty set, because the negative numbers do not have square roots in the set of reals.
$f : \R^2 \to \R$ defined by $f(x, y) = x^2 + y^2.$ {{paragraph break}} The fibers $f^{-1}(\{ a \})$ are concentric circles about the origin, the origin itself, and the empty set (respectively), depending on whether $a > 0, \ a = 0, \text{ or } \ a < 0$ (respectively). (If $a \ge 0,$ then the fiber $f^{-1}(\{ a \})$ is the set of all $(x, y) \in \R^2$ satisfying the equation $x^2 + y^2 = a,$ that is, the origin-centered circle with radius $\sqrt{a}.$ )
If $M$ is a manifold and $\pi : TM \to M$ is the canonical projection from the tangent bundle $TM$ to $M,$ then the fibers of $\pi$ are the tangent spaces $T_x(M) \text{ for } x \in M.$ This is also an example of a fiber bundle.
A quotient group is a homomorphic image.

Properties

{{See also|List of set identities and relations#Functions and sets}}

class=wikitable style="float:right;"

! Counter-examples based on the real numbers $\R,$
$f : \R \to \R$ defined by $x \mapsto x^2,$
showing that equality generally need
not hold for some laws:

File:Image preimage conterexample intersection.gif

File:Image preimage conterexample bf.gif

File:Image preimage conterexample fb.gif

= General =

For every function $f : X \to Y$ and all subsets $A \subseteq X$ and $B \subseteq Y,$ the following properties hold:

class="wikitable"

Image

! Preimage

f(X) \subseteq Y

| $f^{-1}(Y) = X$

f\left(f^{-1}(Y)\right) = f(X)

| $f^{-1}(f(X)) = X$

f\left(f^{-1}(B)\right) \subseteq B

(equal if

B \subseteq f(X);

for instance, if

f

is surjective)See {{harvnb|Halmos|1960|p=31}}See {{harvnb|Munkres|2000|p=19}}

| $f^{-1}(f(A)) \supseteq A$
(equal if $f$ is injective)

f(f^{-1}(B)) = B \cap f(X)

| $\left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)$

f\left(f^{-1}(f(A))\right) = f(A)

| $f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)$

f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing

| $f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)$

f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B

| $f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B$

f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)

| $f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X$

f(X \setminus A) \supseteq f(X) \setminus f(A)

| $f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)$

f\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B

See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.

| $f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)$

f\left(A \cap f^{-1}(B)\right) = f(A) \cap B

| $f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)$

Also:

$f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing$

= Multiple functions =

For functions $f : X \to Y$ and $g : Y \to Z$ with subsets $A \subseteq X$ and $C \subseteq Z,$ the following properties hold:

$(g \circ f)(A) = g(f(A))$
$(g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))$

= Multiple subsets of domain or codomain =

For function $f : X \to Y$ and subsets $A, B \subseteq X$ and $S, T \subseteq Y,$ the following properties hold:

Image ! Preimage
class="wikitable"
$A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)$ \| $S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)$
$f(A \cup B) = f(A) \cup f(B)$ {{harvnb\|Kelley\|1985\|p=[{{Google books\|plainurl=y\|id=-goleb9Ov3oC\|page=85\|text=The image of the union of a family of subsets of X is the union of the images, but, in general, the image of the intersection is not the intersection of the images}} 85]}} \| $f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)$
$f(A \cap B) \subseteq f(A) \cap f(B)$ (equal if $f$ is injectiveSee {{harvnb\|Munkres\|2000\|p=21}}) \| $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$
$f(A \setminus B) \supseteq f(A) \setminus f(B)$ (equal if $f$ is injective) \| $f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)$
$f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)$ (equal if $f$ is injective) \| $f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)$

The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:

$f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)$
$f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f\left(A_s\right)$
$f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}\left(B_s\right)$
$f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}\left(B_s\right)$

(Here, $S$ can be infinite, even uncountably infinite.)

With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice homomorphism (that is, it does not always preserve intersections).

Notes

References

{{Cite book|last=Artin|first=Michael|author-link=Michael Artin|title=Algebra|year=1991|publisher=Prentice Hall|isbn=81-203-0871-9}}
{{cite book|first=T.S.|last=Blyth|title=Lattices and Ordered Algebraic Structures|publisher=Springer|year=2005|isbn=1-85233-905-5}}.
{{Dolecki Mynard Convergence Foundations Of Topology}}
{{cite book|last=Halmos|first=Paul R.|author-link=Paul Halmos|title=Naive set theory|url=https://archive.org/details/naivesettheory0000halm|url-access=registration|series=The University Series in Undergraduate Mathematics|publisher=van Nostrand Company|year=1960|isbn=9780442030643|zbl=0087.04403}}
{{cite book|last1=Kelley|first1=John L.|title=General Topology|edition=2|series=Graduate Texts in Mathematics|volume=27|year=1985|publisher=Birkhäuser|isbn=978-0-387-90125-1}}
{{Munkres Topology|edition=2}}

Category:Basic concepts in set theory

Category:Isomorphism theorems