In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, the kernel of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
(or equivalence kernel
[.]) 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
can tell",
[.] or
* the corresponding
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of the domain.
An unrelated notion is that of the kernel of a non-empty
family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
which by definition is the
intersection of all its elements:
This definition is used in the theory of
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
to classify them as being
free or
principal.
Definition
For the formal definition, let
be a function between two
sets.
Elements
are ''equivalent'' if
and
are
equal, that is, are the same element of
The kernel of
is the equivalence relation thus defined.
The is
The kernel of
is also sometimes denoted by
The kernel of the
empty set,
is typically left undefined.
A family is called and is said to have if its is not empty.
A family is said to be if it is not fixed; that is, if its kernel is the empty set.
Quotients
Like any equivalence relation, the kernel can be
modded out to form a
quotient set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
, and the quotient set is the partition:
This quotient set
is called the ''
coimage In algebra, the coimage of a homomorphism
:f : A \rightarrow B
is the quotient
:\text f = A/\ker(f)
of the domain by the kernel.
The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies.
...
'' of the function
and denoted
(or a variation).
The coimage is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natu ...
(in the set-theoretic sense of a
bijection) to the
image,
specifically, the
equivalence class of
in
(which is an element of
) corresponds to
in
(which is an element of
).
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
In this guise, the kernel may be denoted
(or a variation) and may be defined symbolically as
The study of the properties of this subset can shed light on
Algebraic structures
If
and
are
algebraic structures of some fixed type (such as
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
s,
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
s, or
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s), and if the function
is a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
, then
is a
congruence relation (that is an
equivalence relation that is compatible with the algebraic structure), and the coimage of
is a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of
The bijection between the coimage and the image of
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
in the algebraic sense; this is the most general form of the
first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
.
In topology
If
is a
continuous function between two
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s then the topological properties of
can shed light on the spaces
and
For example, if
is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
then
must be a
closed set.
Conversely, if
is a Hausdorff space and
is a closed set, then the coimage of
if given the
quotient space topology, must also be a Hausdorff space.
A
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
if and only if the kernel of every family of
closed subsets having the
finite intersection property In general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite inters ...
(FIP) is non-empty;
said differently, a space is compact if and only if every family of closed subsets with F.I.P. is
fixed.
See also
*
References
Bibliography
*
*
{{DEFAULTSORT:Kernel (Set Theory)
Abstract algebra
Basic concepts in set theory
Set theory
Topology