Image (mathematics)

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In mathematics, the image of a function is the set of all output values it may produce.

f is a function from domain X to codomain Y. The yellow oval inside Y is the image of f.

More generally, evaluating a given function f at each element of a given subset A of its domain 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 of f, is the set of all elements of the domain that map to the members of B.

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

Definition

The word "image" is used in three related ways. In these definitions, f : XY 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),[1] is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x.

Image of a subset

The image of a subset AX under f, denoted ${\displaystyle f[A]}$ , is the subset of Y which can be defined using set-builder notation as follows:[2][3]

${\displaystyle f[A]=\{f(x)\mid x\in A\}}$

When there is no risk of confusion, ${\displaystyle f[A]}$  is simply written as ${\displaystyle f(A)}$ . This convention is a common one; the intended meaning must be inferred from the context. This makes f[.] a function whose domain is the power set of X (the set of all subsets of X), and whose codomain is the power set of Y. See § 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.[4] This 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×Y, then the set { y∈Y | xRy for some xX } is called the image, or the range, of R. Dually, the set { xX | xRy for some y∈Y } is called the domain of R.

Inverse image

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

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

Other notations include f −1 (B)[5] and f  (B).[6] The inverse image of a singleton, denoted by f −1[{y}] or by f −1[y], is also called the fiber or fibre 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) = x2, 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.

Notation for image and inverse image

The traditional notations used in the previous section can be confusing. An alternative[7] is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

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

Star notation

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

Other terminology

• An alternative notation for f[A] used in mathematical logic and set theory is f "A.[8][9]
• Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f.

Examples

1. f: {1, 2, 3} → {a, b, c, d} defined by ${\displaystyle f(x)=\left\{{\begin{matrix}a,&{\mbox{if }}x=1\\a,&{\mbox{if }}x=2\\c,&{\mbox{if }}x=3.\end{matrix}}\right.}$
The image of the set {2, 3} under f is f({2, 3}) = {a, c}. The image of the function f is {a, c}. The preimage of a is f −1({a}) = {1, 2}. The preimage of {a, b} is also {1, 2}. The preimage of {b, d} is the empty set {}.
2. f: RR defined by f(x) = x2.
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 = {nR | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.
3. f: R2R defined by f(x, y) = x2 + y2.
The fibre f −1({a}) are concentric circles about the origin, the origin itself, and the empty set, depending on whether a > 0, a = 0, or a < 0, respectively. (if a > 0, then the fiber f −1({a}) is the set of all (x, y) ∈ R2 satisfying the equation of the origin-concentric ring x2 + y2 = a.)
4. If M is a manifold and π: TMM is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces Tx(M) for xM. This is also an example of a fiber bundle.
5. A quotient group is a homomorphic image.

Properties

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

Image showing non-equal sets: ${\displaystyle f\left(A_{1}\cap A_{2}\right)\subsetneq f\left(A_{1}\right)\cap f\left(A_{2}\right).}$  The sets ${\displaystyle A_{1}=[-4,2]}$  and ${\displaystyle A_{2}=[-2,4]}$  are shown in blue immediately below the ${\displaystyle x}$ -axis while their intersection ${\displaystyle A_{3}=[-2,2]}$  is shown in green.

${\displaystyle f\left(f^{-1}\left(B_{3}\right)\right)\subsetneq B_{3}.}$

${\displaystyle f^{-1}\left(f\left(A_{4}\right)\right)\supsetneq A_{4}.}$

General

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

Image Preimage
${\displaystyle f(X)\subseteq Y}$  ${\displaystyle f^{-1}(Y)=X}$
${\displaystyle f(f^{-1}(Y))=f(X)}$  ${\displaystyle f^{-1}(f(X))=X}$
${\displaystyle f(f^{-1}(B))\subseteq B}$
(equal if ${\displaystyle B\subseteq f(X)}$ , e.g. ${\displaystyle f}$  is surjective)[10][11]
${\displaystyle f^{-1}(f(A))\supseteq A}$
(equal if ${\displaystyle f}$  is injective)[10][11]
${\displaystyle f(f^{-1}(B))=B\cap f(X)}$  ${\displaystyle (f\vert _{A})^{-1}(B)=A\cap f^{-1}(B)}$
${\displaystyle f(f^{-1}(f(A)))=f(A)}$  ${\displaystyle f^{-1}(f(f^{-1}(B)))=f^{-1}(B)}$
${\displaystyle f(A)=\varnothing \Leftrightarrow A=\varnothing }$  ${\displaystyle f^{-1}(B)=\varnothing \Leftrightarrow B\subseteq Y\setminus f(X)}$
${\displaystyle f(A)\supseteq B\Leftrightarrow \exists C\subseteq A:f(C)=B}$  ${\displaystyle f^{-1}(B)\supseteq A\Leftrightarrow f(A)\subseteq B}$
${\displaystyle f(A)\supseteq f(X\setminus A)\Leftrightarrow f(A)=f(X)}$  ${\displaystyle f^{-1}(B)\supseteq f^{-1}(Y\setminus B)\Leftrightarrow f^{-1}(B)=X}$
${\displaystyle f(X\setminus A)\supseteq f(X)\setminus f(A)}$  ${\displaystyle f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)}$ [10]
${\displaystyle f(A\cup f^{-1}(B))\subseteq f(A)\cup B}$ [12] ${\displaystyle f^{-1}(f(A)\cup B)\supseteq A\cup f^{-1}(B)}$ [12]
${\displaystyle f(A\cap f^{-1}(B))=f(A)\cap B}$ [12] ${\displaystyle f^{-1}(f(A)\cap B)\supseteq A\cap f^{-1}(B)}$ [12]

Also:

• ${\displaystyle f(A)\cap B=\varnothing \Leftrightarrow A\cap f^{-1}(B)=\varnothing }$

Multiple functions

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

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

Multiple subsets of domain or codomain

For function ${\displaystyle f:X\to Y}$  and subsets ${\displaystyle A_{1},A_{2}\subseteq X}$  and ${\displaystyle B_{1},B_{2}\subseteq Y,}$  the following properties hold:

Image Preimage
${\displaystyle A_{1}\subseteq A_{2}\Rightarrow f(A_{1})\subseteq f(A_{2})}$  ${\displaystyle B_{1}\subseteq B_{2}\Rightarrow f^{-1}(B_{1})\subseteq f^{-1}(B_{2})}$
${\displaystyle f(A_{1}\cup A_{2})=f(A_{1})\cup f(A_{2})}$ [12][13] ${\displaystyle f^{-1}(B_{1}\cup B_{2})=f^{-1}(B_{1})\cup f^{-1}(B_{2})}$
${\displaystyle f(A_{1}\cap A_{2})\subseteq f(A_{1})\cap f(A_{2})}$ [12][13]
(equal if ${\displaystyle f}$  is injective[14])
${\displaystyle f^{-1}(B_{1}\cap B_{2})=f^{-1}(B_{1})\cap f^{-1}(B_{2})}$
${\displaystyle f(A_{1}\setminus A_{2})\supseteq f(A_{1})\setminus f(A_{2})}$ [12]
(equal if ${\displaystyle f}$  is injective[14])
${\displaystyle f^{-1}(B_{1}\setminus B_{2})=f^{-1}(B_{1})\setminus f^{-1}(B_{2})}$ [12]
${\displaystyle f(A_{1}\triangle A_{2})\supseteq f(A_{1})\triangle f(A_{2})}$
(equal if ${\displaystyle f}$  is injective)
${\displaystyle f^{-1}(B_{1}\triangle B_{2})=f^{-1}(B_{1})\triangle f^{-1}(B_{2})}$

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:

• ${\displaystyle f\left(\bigcup _{s\in S}A_{s}\right)=\bigcup _{s\in S}f(A_{s})}$
• ${\displaystyle f\left(\bigcap _{s\in S}A_{s}\right)\subseteq \bigcap _{s\in S}f(A_{s})}$
• ${\displaystyle f^{-1}\left(\bigcup _{s\in S}B_{s}\right)=\bigcup _{s\in S}f^{-1}(B_{s})}$
• ${\displaystyle f^{-1}\left(\bigcap _{s\in S}B_{s}\right)=\bigcap _{s\in S}f^{-1}(B_{s})}$

(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 (i.e., it does not always preserve intersections).

Notes

1. ^ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
2. ^ "5.4: Onto Functions and Images/Preimages of Sets". Mathematics LibreTexts. 2019-11-05. Retrieved 2020-08-28.
3. ^ Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand. Here: Sect.8
4. ^ Weisstein, Eric W. "Image". mathworld.wolfram.com. Retrieved 2020-08-28.
5. ^ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-28.
6. ^ Dolecki & Mynard 2016, pp. 4–5.
7. ^ Blyth 2005, p. 5.
8. ^ Jean E. Rubin (1967). Set Theory for the Mathematician. Holden-Day. p. xix. ASIN B0006BQH7S.
9. ^ M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
10. ^ a b c See Halmos 1960, p. 39
11. ^ a b See Munkres 2000, p. 19
12. See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
13. ^ a b Kelley 1985, p. 85
14. ^ a b See Munkres 2000, p. 21