# One

would like equivalence relation in mathematics in many connections of objects, which resemble each other in certain aspects, when equivalently regard. A formalizing of the minimum requirements to such a equivalence term is the term of the equivalence relation.

For example each term is, as thoseEquality of certain characteristics to be defined knows, an equivalence relation:

• the equal power of finite quantities: two finite quantities mean alike, if they have the same number of elements;
• the congruence in geometry: two triangles are congruent, if them the same side lengthshave;
• the similarity in geometry: two triangles are similar, if they have the same internal angles;
• the congruence in the elementary number theory: two whole numbers are called congruently modulo 5, if them with integral division by 5 the same remainder leave;
• the equality.

Often such terms develop also from the existence of illustrations or other connections with certain characteristics:

• two finite quantities are alike, if there is a bijektive illustration between them;
• two triangles are congruent, if them througha congruence illustration come out apart;
• two triangles are similar, if they come out apart by a similarity illustration;
• two whole numbers are congruently modulo 5, if they come out apart by addition of an integral multiples of 5.

The word “equivalent” stands in the following forone of these relations between two objects; that two objects [itex] A< /math> and [itex] b< /math> equivalent are, are through [itex] A \ sim b< /math> symbolized.

All these terms have the following three characteristics:

Each object is equivalent to itself.
• Symmetry:[itex] A \ sim b \ quad \ iff \ quad b \ sim A< /math>
If [itex] A< /math> too [itex] b< /math> equivalent is, then is also [itex] b< /math> equivalent too [itex] A< /math> (and in reverse).
• Transitivity: [itex] A \ sim b \ \ mathrm {and} \ b \ sim C \ quad \ Longrightarrow \ quad A \ sim C< /math>
If [itex] A< /math> too [itex] b< /math> equivalent and [itex] b< /math> too [itex] C< /math> equivalent is, thenis [itex] A< /math> equivalent too [itex] C< /math>.

Each relationship between objects, which has these characteristics, is called an equivalence relation.

The equivalence class of an object [itex] A< /math> the class of the objects is, those equivalent too [itex] A< /math> are.

In connection with equivalence relations there are three differentAspects:

• individual objects: Are two objects equivalent or not?
• individual equivalence classes: How does the whole of the objects look, which are equivalent to a given?
• the quantity of the equivalence classes: As the whole of all objects looks, if one equivalent objects asdirectly regards?

## descriptive example

an example from the agriculture is the imported termsclarify in front. We regard the quantity of all utilizable animals in a agricultural enterprise. We define now a relation: We say, two animals stand in relation to each other, if they are from the same kind. The cow Erna for example stands with the oxBruno in relation, but not with the chicken Betti. This relation is an equivalence relation: Each animal is of the same kind as it (= “reflexiv”). If an animal of the same kind is as the other one, then the other one is also ofthe same kind as one (= “symmetrically”). If Erna and Lisa of the same kind are and Lisa and Bruno of the same kind, then are Erna and Bruno of the same kind (e.g. Cattle; = “transitiv”). An equivalence class exists here thus outthe animals of a kind. For example chickens form an equivalence class and the cattle another equivalence class.

## formal definition of an equivalence relation

an equivalence relation on a quantity of M is a subset [itex] R \ subseteq M \ times M< /math>, the which followingConditions fulfills:

• Reflexivity: For all [itex] A \ in M< /math> is [itex] (A, A) \ in R< /math>.
• Symmetry: For all [itex] A, b \ in M< /math>, for [itex] (A, b) \ in R< /math> applies, is [itex] (b, A) \ in R< /math>.
• Transitivity: Math <A>, b, are C \ in M< /math> in such a manner that [itex] (A, b) \ in R< /math> and [itex] (b, C) \ in R< /math> applies,so is also [itex] (A, C) \ in R< /math>.

Usually one writes

< math> \ sim_Rb /math< to A> or simply [itex] A \ sim b< /math> instead of [itex] (A, b) \ in R,< /math>

and then take this demand exactly the form specified in the introduction.

## equivalence classes

is [itex] R< /math> an equivalence relation on oneMath <M> /math<,> then one calls quantity for an element [itex] A \ in M< /math> the subset

< math> [A] _R = \ {x \ in M \ avoided x \ sim_R A \} \ subseteq M< /math>

[itex] R< /math> - equivalence class of [itex] A< /math>. It is clear from the context that equivalence classes relative [itex] R< /math> , leaves one is educated the additive“[itex] R< /math> - “away. Other ways of writing are

Elements of an equivalence class are called their representatives or representatives.

Each element of [itex] M< /math> is contained in exactly one equivalence class. The equivalence classes to two elements are either alike or disjunkt, firstexactly if the elements are equivalent:

[itex] [A] = [b] \ quad \ iff \ quad A \ sim b \ quad \ iff \ quad A \ in [b] \ quad \ iff \ quad b \ in [A]. [/itex]

The equivalence classes form a partition of [itex] M< /math>.

## the quantity of the equivalence classes

the quantity of the equivalence classes (sometimes also factor quantity called) is

[itex] m {\ sim}: = \ {[A] \ A avoided\ in M \}. [/itex]

The Kardinalität [itex] \ left|M {\ sim} \ right|[/itex] sometimes becomes also as the index of the equivalence relation [itex] R< /math> designated. (A special case is the index of a sub-group.)

the quantity of the equivalence classes is that quantity, which develops, if one makes equivalent elements “alike”. See also Identification abstraction.

There is a canonical surjektive illustration

< math> M \ tons of m {\ sim}, \ quad m \ mapsto [m]. [/itex]

## examples

• grades: the underlying quantity of M is the quantity of all pupils on a school; two pupils are equivalent, if they go into the same class.
• Equivalence classa pupil is its class.
• The quantity of the equivalence classes is the quantity of the classes.
• Equality: on any quantity [itex] M< /math> two elements are equivalent, if they are alike:
[itex] m \ sim n \ quad: \ Longleftrightarrow \ quad m=n< /math>
• Equivalence class of an element [itex] m< /math> is the einelementige quantity[itex] \ {m \}< /math>.
• The quantity of the equivalence classes is the quantity of the einelementigen subsets of [itex] M< /math>; the illustration [itex] M \ tons of m {\ sim}< /math> is a Bijektion.
• Equivalence class of a whole number [itex] k< /math> the remainder class in such a way specified is
[itex] \ without k=k+5 \ mathbb Z= \ {k+5z \ avoided z \ in \ mathbb Z \} = \ {\ ldots, k-10, k-5, k, k+5, k+10, \ ldots \}. [/itex]
• The quantity of the equivalence classesthe remainder class ring is
[itex] \ mathbb Z/5 \ mathbb Z= \ {\ bar0, \ bar1, \ bar2, \ bar3, \ bar4 \}. [/itex]
• Breaks: It is [itex] M: = \ {(z, n) \ in \ mathbb Z^2 \ n avoided \ not=0 \}< /math> the quantity of the pairs of whole numbers, whose second entry of zero is different. Two pairs [itex] (z_1, n_1)< /math> and [itex] (z_2, n_2)< /math> are equivalent to mean, if applies:
[itex] \ frac {z_1} {n_1} = \ frac {z_2} {n_2}. [/itex]
• The equivalence class onePair [itex] (z, n)< /math> /math consists of all pairs (counter, denominator) for break representations of the rational <number> math \ frac< zn>.
• The quantity of the equivalence classes becomes through
< math> m {\ sim} \ tons \ mathbb Q, \ quad [(z, n)]\ mapsto \ frac zn< /math>
bijektiv on the quantity of the rational numbers shown. An important point is here the well-definedness: if [itex] [(z_1, n_1)]= [(z_2, n_2)][/itex] applies, then the picture is according to the above regulation
on the one hand [itex] \ frac {z_1} {n_1}< /math>, on the other hand [itex] \ frac {z_2} {n_2}< /math>.
The equivalence relation was however straight in such a way selected that equal these two rational numbers are.

## universal characteristic

is [itex] M< /math> a quantity, [itex] R< /math>an equivalence relation on [itex] M< /math> and [itex] N< /math> a further quantity, then the illustration math <M> arrange \ for tons of m {\ sim_R}< /math> a Bijektion between the following quantities:

• Illustrations [itex] M \ tons of N< /math>, with those [itex] R< /math> - equivalent elements the same picture have

and

• illustrations [itex] m {\ sim_R} \ tons of N.< /math>

## further equivalence terms

special ones Meaning is attached to equivalence relations, which are compatible with an algebraic structure on a quantity; the principal interest applies here for the quantity of the equivalence classes, which the algebraic structure “inherits”:

The following equivalence terms develop from the demand that a pair of illustrations with certain characteristics between two objects exists, which “more or less” inversely to each other are:

further ones Examples of equivalence relations: