Kernel (set theory)

Kernel (set theory)

In mathematics, 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", or
*the corresponding partition of the domain.

Note that there are several other meanings of the word "kernel" in mathematics; see kernel (mathematics) for these.

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.

The kernel, in the equivalence-relation sense, may be denoted "="f"" (or a variation) and may be defined symbolically as

: (x =_{f} y); :Longleftrightarrow; (f(x) = f(y)). !

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: f(x)=f(w) , ight} : x in X , ight}.

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

: mathop{mathrm{ker f := {(x,x') mid f(x) = f(x')}mbox{.} !

But this is not useful merely as a formalisation in set theory!In fact, the study of the properties of this subset can shed important light on the function in question.We give here two examples.

First, 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" will be a subalgebra of the direct product "X" × "X". Subalgebras of "X" × "X" that are also equivalence relations (called "congruence relations") are important in abstract algebra, because they define the most general notion of quotient algebra. Thus the coimage of "f" is a quotient algebra of "X" much as the image of "f" is a subalgebra of "Y"; and the bijection between them becomes an isomorphism in the algebraic sense as well (this is the most general form of the first isomorphism theorem in algebra). The use of kernels in this context is discussed further in the article Kernel (algebra).

Secondly, 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.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Kernel (mathematics) — In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element (such as zero or zero vector), as in kernel of a linear… …   Wikipedia

  • Kernel (algebra) — In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also… …   Wikipedia

  • Kernel density estimation — of 100 normally distributed random numbers using different smoothing bandwidths. In statistics, kernel density estimation is a non parametric way of estimating the probability density function of a random variable. Kernel density estimation is a… …   Wikipedia

  • Kernel methods — (KMs) are a class of algorithms for pattern analysis, whose best known elementis the Support Vector Machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal… …   Wikipedia

  • Kernel (computing) — A kernel connects the application software to the hardware of a computer In computing, the kernel is the main component of most computer operating systems; it is a bridge between applications and the actual data processing done at the hardware… …   Wikipedia

  • Kernel (matrix) — In linear algebra, the kernel or null space (also nullspace) of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n dimensional Euclidean space.[1] The dimension… …   Wikipedia

  • theory — A reasoned explanation of known facts or phenomena that serves as a basis of investigation by which to seek the truth. SEE ALSO: hypothesis, postulate. [G. theoria, a beholding, speculation, t., fr. theoros, a beholder] adsorption t. of narcosis… …   Medical dictionary

  • Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… …   Wikipedia

  • Limit (category theory) — In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint… …   Wikipedia

  • Multivariate kernel density estimation — Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density… …   Wikipedia