Image (mathematics)

File:FunctionMappingPersonToFavoriteFood.png

For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.

{{#invoke:other uses|otheruses}}

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

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

The image of the function <math>f</math> is the set of all output values it may produce, that is, the image of <math>X</math>. The preimage of <math>f</math> is the preimage of the codomain <math>Y</math>. Because it always equals <math>X</math> (the domain of <math>f</math>), it is rarely used.

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

Definition

File:ImagePreimageOfElement.png

<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

<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

<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 from the 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 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 Template:Em or Template:Em 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 Template:Em if there exists Template:Em <math>x</math> in the function's domain such that <math>f(x) \in S.</math> However, Template:Em and Template:Em means that <math>f(x) \in S</math> for Template:Em 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 Template:AnchorTemplate:Em 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>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book 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 subsets of <math>S.</math> See Template:Section link 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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>

Inverse image

Template:Redirect Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The preimage or inverse image of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by <math display="block">f^{-1}[ B ] = \{ x \in X \,:\, f(x) \in B \}.</math>

Other notations include <math>f^{-1}(B)</math> and <math>f^{-}(B).</math>Template:Sfn The inverse image of a singleton set, denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}(y),</math> is also called the fiber or fiber over <math>y</math> or the level set of <math>y.</math> The set of all the fibers over the elements of <math>Y</math> is a family of sets indexed by <math>Y.</math>

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

Notation for image and inverse image

The traditional notations used in the previous section do not distinguish the original function <math>f : X \to Y</math> from the image-of-sets function <math>f : \mathcal{P}(X) \to \mathcal{P}(Y)</math>; 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 alternativeTemplate:Sfn is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

• <math>f^\rightarrow : \mathcal{P}(X) \to \mathcal{P}(Y)</math> with <math>f^\rightarrow(A) = \{ f(a)\;|\; a \in A\}</math>
• <math>f^\leftarrow  : \mathcal{P}(Y) \to \mathcal{P}(X)</math> with <math>f^\leftarrow(B) = \{ a \in X \;|\; f(a) \in B\}</math>

Star notation

• <math>f_\star : \mathcal{P}(X) \to \mathcal{P}(Y)</math> instead of <math>f^\rightarrow</math>
• <math>f^\star : \mathcal{P}(Y) \to \mathcal{P}(X)</math> instead of <math>f^\leftarrow</math>

Other terminology

• An alternative notation for <math>f[A]</math> used in mathematical logic and set theory is <math>f\,A.</math><ref>Template:Cite book</ref><ref>M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2</ref>
• Some texts refer to the image of <math>f</math> as the range of <math>f,</math><ref>Template:Cite book</ref> but this usage should be avoided because the word "range" is also commonly used to mean the codomain of <math>f.</math>

Examples

1. <math>f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}</math> defined by <math>

   \left\{\begin{matrix}
     1 \mapsto a, \\
     2 \mapsto a, \\
     3 \mapsto c.
   \end{matrix}\right.
 </math>Template:Paragraph break The image of the set <math>\{ 2, 3 \}</math> under <math>f</math> is <math>f(\{ 2, 3 \}) = \{ a, c \}.</math> The image of the function <math>f</math> is <math>\{ a, c \}.</math> The preimage of <math>a</math> is <math>f^{-1}(\{ a \}) = \{ 1, 2 \}.</math> The preimage of <math>\{ a, b \}</math> is also <math>f^{-1}(\{ a, b \}) = \{ 1, 2 \}.</math> The preimage of <math>\{ b, d \}</math> under <math>f</math> is the empty set <math>\{ \ \} = \emptyset.</math>

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

Properties

Counter-examples based on the real numbers <math>\R,</math> <math>f : \R \to \R</math> defined by <math>x \mapsto x^2,</math> showing that equality generally need not hold for some laws:
File:Image preimage conterexample intersection.gif Image showing non-equal sets: <math>f\left(A \cap B\right) \subsetneq f(A) \cap f(B).</math> The sets <math>A = [-4, 2]</math> and <math>B = [-2, 4]</math> are shown in Template:Ifsubst style="color:blue">blue immediately below the <math>x</math>-axis while their intersection <math>A_3 = [-2, 2]</math> is shown in Template:Ifsubst style="color:green">green.
File:Image preimage conterexample bf.gif <math>f\left(f^{-1}\left(B_3\right)\right) \subsetneq B_3.</math>
File:Image preimage conterexample fb.gif <math>f^{-1}\left(f\left(A_4\right)\right) \supsetneq A_4.</math>

General

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

Image	Preimage
<math>f(X) \subseteq Y</math>	<math>f^{-1}(Y) = X</math>
<math>f\left(f^{-1}(Y)\right) = f(X)</math>	<math>f^{-1}(f(X)) = X</math>
<math>f\left(f^{-1}(B)\right) \subseteq B</math> (equal if <math>B \subseteq f(X);</math> for instance, if <math>f</math> is surjective)<ref name="halmos-1960-p31">See Template:Harvnb</ref><ref name="munkres-2000-p19">See Template:Harvnb</ref>	<math>f^{-1}(f(A)) \supseteq A</math> (equal if <math>f</math> is injective)<ref name="halmos-1960-p31"/><ref name="munkres-2000-p19" />
<math>f(f^{-1}(B)) = B \cap f(X)</math>	<math>\left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)</math>
<math>f\left(f^{-1}(f(A))\right) = f(A)</math>	<math>f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)</math>
<math>f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing</math>	<math>f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)</math>
<math>f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B</math>	<math>f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B</math>
<math>f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)</math>	<math>f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X</math>
<math>f(X \setminus A) \supseteq f(X) \setminus f(A)</math>	<math>f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)</math><ref name="halmos-1960-p31" />
<math>f\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B</math><ref name="lee-2010-p388">See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.</ref>	<math>f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)</math><ref name="lee-2010-p388" />
<math>f\left(A \cap f^{-1}(B)\right) = f(A) \cap B</math><ref name="lee-2010-p388" />	<math>f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)</math><ref name="lee-2010-p388" />

Also:

• <math>f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing</math>

Multiple functions

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

• <math>(g \circ f)(A) = g(f(A))</math>
• <math>(g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))</math>

Multiple subsets of domain or codomain

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

Image	Preimage
<math>A \subseteq B \,\text{ implies }\, f(A) \subseteq f(B)</math>	<math>S \subseteq T \,\text{ implies }\, f^{-1}(S) \subseteq f^{-1}(T)</math>
<math>f(A \cup B) = f(A) \cup f(B)</math><ref name="lee-2010-p388" /><ref name="kelley-1985">Template:Harvnb</ref>	<math>f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)</math>
<math>f(A \cap B) \subseteq f(A) \cap f(B)</math><ref name="lee-2010-p388" /><ref name="kelley-1985" /> (equal if <math>f</math> is injective<ref name="munkres-2000-p21">See Template:Harvnb</ref>)	<math>f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)</math>
<math>f(A \setminus B) \supseteq f(A) \setminus f(B)</math><ref name="lee-2010-p388" /> (equal if <math>f</math> is injective<ref name="munkres-2000-p21" />)	<math>f^{-1}(S \setminus T) = f^{-1}(S) \setminus f^{-1}(T)</math><ref name="lee-2010-p388" />
<math>f\left(A \triangle B\right) \supseteq f(A) \triangle f(B)</math> (equal if <math>f</math> is injective)	<math>f^{-1}\left(S \triangle T\right) = f^{-1}(S) \triangle f^{-1}(T)</math>

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:

• <math>f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)</math>
• <math>f\left(\bigcap_{s\in S}A_s\right) \subseteq \bigcap_{s\in S} f\left(A_s\right)</math>
• <math>f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}\left(B_s\right)</math>
• <math>f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}\left(B_s\right)</math>

(Here, <math>S</math> 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).

See also

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References

• Template:Cite book
• Template:Cite book.
• Template:Dolecki Mynard Convergence Foundations Of Topology
• Template:Cite book
• Template:Cite book
• Template:Munkres Topology

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