In
mathematics, when the elements of some
set have a notion of equivalence (formalized as an
equivalence relation), then one may naturally split the set
into equivalence classes. These equivalence classes are constructed so that elements
and
belong to the same equivalence class
if, and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
, they are equivalent.
Formally, given a set
and an equivalence relation
on
the of an element
in
denoted by
is the set
of elements which are equivalent to
It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a
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
This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of
by
and is denoted by
When the set
has some structure (such as a
group operation or a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
) and the equivalence relation
is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include
quotient spaces in linear algebra,
quotient spaces in topology,
quotient groups,
homogeneous spaces,
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
s,
quotient monoid
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy ...
s, and
quotient categories.
Examples
* If
is the set of all cars, and
is the
equivalence relation "has the same color as", then one particular equivalence class would consist of all green cars, and
could be naturally identified with the set of all car colors.
* Let
be the set of all rectangles in a plane, and
the equivalence relation "has the same area as", then for each positive real number
there will be an equivalence class of all the rectangles that have area
* Consider the
modulo 2 equivalence relation on the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s,
such that
if and only if their difference
is an
even number. This relation gives rise to exactly two equivalence classes: one class consists of all even numbers, and the other class consists of all odd numbers. Using square brackets around one member of the class to denote an equivalence class under this relation,
and