Kernel (set theory)
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.
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
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. 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.
- Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, 301, CRC Press, pp. 14–16, ISBN 9781439851296.