Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce.
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.
DefinitionEdit
The word "image" is used in three related ways. In these definitions, f : X → Y is a function from the set X to the set Y.
Image of an elementEdit
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 subsetEdit
The image of a subset A ⊆ X under f, denoted , is the subset of Y which can be defined using setbuilder notation as follows:^{[2]}
When there is no risk of confusion, is simply written as . 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 functionEdit
The image of a function is the image of its entire domain, also known as the range of the function.^{[3]}
Generalization to binary relationsEdit
If R is an arbitrary binary relation on X×Y, then the set { y∈Y  xRy for some x∈X } is called the image, or the range, of R. Dually, the set { x∈X  xRy for some y∈Y } is called the domain of R.
Inverse imageEdit
Let f be a function from X to Y. The preimage or inverse image of a set B ⊆ Y under f, denoted by , is the subset of X defined by
Other notations include f ^{−1} (B)^{[4]} and f ^{−} (B).^{[5]} The inverse image of a singleton, denoted by f^{ −1}[{y}] or by f^{ −1}[y], is also called the 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}.
Notation for image and inverse imageEdit
The traditional notations used in the previous section can be confusing. An alternative^{[6]} is to give explicit names for the image and preimage as functions between power sets:
Arrow notationEdit
 with
 with
Star notationEdit
 instead of
 instead of
Other terminologyEdit
 An alternative notation for f[A] used in mathematical logic and set theory is f "A.^{[7]}^{[8]}
 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.
ExamplesEdit
 f: {1, 2, 3} → {a, b, c, d} defined by 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 {}.
 f: R → R defined by f(x) = x^{2}. The image of {−2, 3} under f is f({−2, 3}) = {4, 9}, and the image of f is R^{+}. The preimage of {4, 9} under f is f^{ −1}({4, 9}) = {−3, −2, 2, 3}. The preimage of set N = {n ∈ 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} → R defined by f(x, y) = x^{2} + y^{2}. The fibres 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 M is a manifold and π: TM → M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces T_{x}(M) for x∈M. This is also an example of a fiber bundle.
 A quotient group is a homomorphic image.
PropertiesEdit
Counterexamples based on f:ℝ→ℝ, x↦x^{2}, showing that equality generally need not hold for some laws: 

GeneralEdit
For every function and all subsets and , the following properties hold:
Image  Preimage 

(equal if , e.g. is surjective)^{[9]}^{[10]} 
(equal if is injective)^{[9]}^{[10]} 
^{[9]}  
^{[11]}  ^{[11]} 
^{[11]}  ^{[11]} 
Also:
Multiple functionsEdit
For functions and with subsets and , the following properties hold:
Multiple subsets of domain or codomainEdit
For function and subsets and , the following properties hold:
Image  Preimage 

^{[11]}^{[12]}  
^{[11]}^{[12]} (equal if is injective^{[13]}) 

^{[11]} (equal if is injective^{[13]}) 
^{[11]} 
(equal if is injective) 
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:
(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).
See alsoEdit
NotesEdit
 ^ "Compendium of Mathematical Symbols". Math Vault. 20200301. Retrieved 20200828.
 ^ "5.4: Onto Functions and Images/Preimages of Sets". Mathematics LibreTexts. 20191105. Retrieved 20200828.
 ^ Weisstein, Eric W. "Image". mathworld.wolfram.com. Retrieved 20200828.
 ^ "Comprehensive List of Algebra Symbols". Math Vault. 20200325. Retrieved 20200828.
 ^ Dolecki & Mynard 2016, pp. 45.
 ^ Blyth 2005, p. 5.
 ^ Jean E. Rubin (1967). Set Theory for the Mathematician. HoldenDay. p. xix. ASIN B0006BQH7S.
 ^ M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
 ^ ^{a} ^{b} ^{c} See Halmos 1960, p. 39
 ^ ^{a} ^{b} See Munkres 2000, p. 19
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
 ^ ^{a} ^{b} Kelley 1985, p. 85
 ^ ^{a} ^{b} See Munkres 2000, p. 21
ReferencesEdit
 Artin, Michael (1991). Algebra. Prentice Hall. ISBN 8120308719.
 Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1852339055..
 Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 9789814571524. OCLC 945169917.
 Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403.
 Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics. 27 (2 ed.). Birkhäuser. ISBN 9780387901251.
 Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 9780131816299. OCLC 42683260.
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