In set theory, the kernel of a function f 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",[1] or
the corresponding partition of the domain.
Definition
For the formal definition, let X and Y be sets and let f be a function from X to Y. Elements x1 and x2 of X are equivalent if f(x1) and f(x2) are equal, i.e. are the same element of Y. The kernel of f is the equivalence relation thus defined.[1]
Quotients
Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:
\( \left\{\,\left\{\,w\in X\mid f(x)=f(w)\,\right\}\mid x\in X\,\right\}. \)
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).
As a subset of the square
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
\( \operatorname {ker}f:=\{(x,x')\mid f(x)=f(x')\}. \) [1]
The study of the properties of this subset can shed light on f.
In algebraic structures
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.[1] 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).
In topological spaces
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.
References
Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, 301, CRC Press, pp. 14–16, ISBN 9781439851296.
Sources
Awodey, Steve (2010) [2006]. Category Theory (PDF). Oxford Logic Guides. 49 (2nd ed.). Oxford University Press. ISBN 978-0-19-923718-0.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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