# Image (mathematics)

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

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* ". 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.

## Contents

## 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 *f*(*x*) = *y* (the value of *f* when applied to *x*) is the image of *x* under *f*. *y* is alternatively known as the output of *f* for argument *x*.

### Image of a subsetEdit

The image of a subset *A* ⊆ *X* under *f* is the subset *f*[*A*] ⊆ *Y* defined by (using set-builder notation):

When there is no risk of confusion, *f*[*A*] is simply written as *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.

### Image of a functionEdit

The *image* of a function is the image of its entire domain.

### Generalization to binary relationsEdit

If *R* is an arbitrary binary relation on *X*×*Y*, 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* is the subset of *X* defined by

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, denote *f*^{ −1}[*B*] by *f*^{ −1}(*B*), and think of *f*^{ −1} 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. The notation coincides with the usual one, though, for bijections, in the sense 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^{[1]} is to give explicit names for the image and preimage as functions between powersets:

### 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*.^{[2]}^{[3]} - 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

Counter-examples based onf:ℝ→ℝ, x↦x^{2}, showingthat equality generally need not hold for some laws: |
---|

For every function *f* : *X* → *Y*, all subsets *A*, *A*_{1}, and *A*_{2} of *X* and all subsets *B*, *B*_{1}, and *B*_{2} of *Y*, the following properties hold:

*f*(*A*_{1}∪*A*_{2}) =*f*(*A*_{1}) ∪*f*(*A*_{2})^{[4]}*f*(*A*_{1}∩*A*_{2}) ⊆*f*(*A*_{1}) ∩*f*(*A*_{2})^{[4]}*f*(*A*∩*f*^{ −1}(*B*)) =*f*(*A*) ∩*B**f*^{ −1}(*B*_{1}∪*B*_{2}) =*f*^{ −1}(*B*_{1}) ∪*f*^{ −1}(*B*_{2})*f*^{ −1}(*B*_{1}∩*B*_{2}) =*f*^{ −1}(*B*_{1}) ∩*f*^{ −1}(*B*_{2})*f*(*A*) = ∅ ⇔*A*= ∅*f*^{ −1}(*B*) = ∅ ⇔*B*⊆ (*f*(*X*))^{C}*f*(*A*) ∩*B*= ∅ ⇔*A*∩*f*^{ −1}(*B*) = ∅*f*(*A*) ⊆*B*⇔*A*⊆*f*^{ −1}(*B*)*B*⊆*f*(*A*) ⇔*C*⊆*A*(*f*(*C*) =*B*)*f*(*f*^{ −1}(*B*)) ⊆*B*^{[5]}*f*^{ −1}(*f*(*A*)) ⊇*A*^{[6]}*f*(*f*^{ −1}(*B*)) = B ∩*f*(*X*)*f*^{ −1}(*f*(*X*)) =*X**A*_{1}⊆*A*_{2}⇒*f*(*A*_{1}) ⊆*f*(*A*_{2})*B*_{1}⊆*B*_{2}⇒*f*^{ −1}(*B*_{1}) ⊆*f*^{ −1}(*B*_{2})*f*^{ −1}(*B*^{C}) = (*f*^{ −1}(*B*))^{C}- (
*f*|_{A})^{−1}(*B*) =*A*∩*f*^{ −1}(*B*).

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, by the above the inverse image function is a lattice homomorphism while the image function is only a semilattice homomorphism (it does not always preserve intersections).

## See alsoEdit

## NotesEdit

**^**Blyth 2005, p. 5**^**Jean E. Rubin (1967).*Set Theory for the Mathematician*. Holden-Day. 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}Kelley (1985), p. 85 **^**Equality holds if*B*is a subset of*f*(*X*) or, in particular, if*f*is surjective. See Munkres, J.. Topology (2000), p. 19.**^**Equality holds if*f*is injective. See Munkres, J.. Topology (2000), p. 19.

## ReferencesEdit

- Artin, Michael (1991).
*Algebra*. Prentice Hall. ISBN 81-203-0871-9 - T.S. Blyth,
*Lattices and Ordered Algebraic Structures*, Springer, 2005, ISBN 1-85233-905-5. - Munkres, James R. (2000).
*Topology*(2 ed.). Prentice Hall. ISBN 978-0-13-181629-9. - Kelley, John L. (1985).
*General Topology*. Graduate Texts in Mathematics.**27**(2 ed.). Birkhäuser. ISBN 978-0-387-90125-1.

*This article incorporates material from Fibre on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*