Equivalence class

In mathematics, when the elements of some set

S

{\displaystyle S}

have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set

S

{\displaystyle S}

into equivalence classes. These equivalence classes are constructed so that elements

a

{\displaystyle a}

and

b

{\displaystyle b}

belong to the same equivalence class if, and only if, they are equivalent.

Formally, given a set

S

{\displaystyle S}

and an equivalence relation

∼

{\displaystyle \sim }

on

S

,

{\displaystyle S,}

the equivalence class of an element

a

{\displaystyle a}

in

S

{\displaystyle S}

is denoted

[

a

]

{\displaystyle [a]}

or, equivalently,

[

a

]

∼

{\displaystyle [a]_{\sim }}

to emphasize its equivalence relation

∼

{\displaystyle \sim }

, and is defined as the set of all elements in

S

{\displaystyle S}

with which

a

{\displaystyle a}

is

∼

{\displaystyle \sim }

-related. The definition of equivalence relations implies that the equivalence classes form a partition of

S

,

{\displaystyle S,}

meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of

S

{\displaystyle S}

by

∼

,

{\displaystyle \sim ,}

and is denoted by

S

/

∼

.

{\displaystyle S/{\sim }.}

When the set

S

{\displaystyle S}

has some structure (such as a group operation or a topology) and the equivalence relation

∼

{\displaystyle \sim }

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 rings, quotient monoids, and quotient categories.

Definition and notation

An equivalence relation on a set

X

{\displaystyle X}

is a binary relation

∼

{\displaystyle \sim }

on

X

{\displaystyle X}

satisfying the three properties:

a

∼

a

{\displaystyle a\sim a}

for all

a

∈

X

{\displaystyle a\in X}

(reflexivity),

a

∼

b

{\displaystyle a\sim b}

implies

b

∼

a

{\displaystyle b\sim a}

for all

a

,

b

∈

X

{\displaystyle a,b\in X}

(symmetry),

if

a

∼

b

{\displaystyle a\sim b}

and

b

∼

c

{\displaystyle b\sim c}

then

a

∼

c

{\displaystyle a\sim c}

for all

a

,

b

,

c

∈

X

{\displaystyle a,b,c\in X}

(transitivity).

The equivalence class of an element

a

{\displaystyle a}

is defined as

[

a

]

=

{

x

∈

X

:

a

∼

x

}

.

{\displaystyle [a]=\{x\in X:a\sim x\}.}

The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although some equivalence classes are not sets but proper classes.