Inverse Image

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## Definition

### Image of an element

### Image of a subset

### Image of a function

### Generalization to binary relations

## Inverse image

## Notation for image and inverse image

### Arrow notation

### Star notation

### Other terminology

## Examples

## Properties

## See also

## Notes

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Inverse Image

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* ". 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.

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*.

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*.

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.

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

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*.

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}.

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:

- with
- with

- instead of
- instead of

- 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*.

*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*?*:*TM*->*M*is the canonical projection from the tangent bundle*TM*to*M*, then the*fibres*of*?*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.

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).

- Range (mathematics)
- Bijection, injection and surjection
- Kernel of a function
- Image (category theory)
- Set inversion

**^**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.

- 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.*

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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