Kernel of A Function

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## Definition

## Quotients

## As a subset of the square

## In algebraic structures

## In topological spaces

## References

## Sources

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Kernel of A Function

In set theory, the **kernel** of a function *f* (or **equivalence kernel**^{[1]}) may be taken to be either

- the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function
*f*can tell",^{[2]}or - the corresponding partition of the domain.

For the formal definition, let *X* and *Y* be sets and let *f* be a function from *X* to *Y*.
Elements *x*_{1} and *x*_{2} of *X* are *equivalent* if *f*(*x*_{1}) and *f*(*x*_{2}) are equal, i.e. are the same element of *Y*.
The kernel of *f* is the equivalence relation thus defined.^{[2]}

Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:

This quotient set *X* /=_{f} is called the *coimage* of the function *f*, and denoted coim *f* (or a variation).
The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, im *f*; specifically, the equivalence class of *x* in *X* (which is an element of coim *f*) corresponds to *f*(*x*) in *Y* (which is an element of im *f*).

Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product *X* × *X*.
In this guise, the kernel may be denoted ker *f* (or a variation) and may be defined symbolically as

- .
^{[2]}

The study of the properties of this subset can shed light on *f*.

If *X* and *Y* are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function *f* from *X* to *Y* is a homomorphism, then ker *f* is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of *f* is a quotient of *X*.^{[2]}
The bijection between the coimage and the image of *f* is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem. See also Kernel (algebra).

If *X* and *Y* are topological spaces and *f* is a continuous function between them, then the topological properties of ker *f* can shed light on the spaces *X* and *Y*.
For example, if *Y* is a Hausdorff space, then ker *f* must be a closed set.
Conversely, if *X* is a Hausdorff space and ker *f* is a closed set, then the coimage of *f*, if given the quotient space topology, must also be a Hausdorff space.

**^**Mac Lane, Saunders; Birkhoff, Garrett (1999),*Algebra*, Chelsea Publishing Company, p. 33, ISBN 0821816462.- ^
^{a}^{b}^{c}^{d}Bergman, Clifford (2011),*Universal Algebra: Fundamentals and Selected Topics*, Pure and Applied Mathematics,**301**, CRC Press, pp. 14-16, ISBN 9781439851296.

- Awodey, Steve (2010) [2006].
*Category Theory*. Oxford Logic Guides.**49**(2nd ed.). Oxford University Press. ISBN 978-0-19-923718-0.

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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