A relation 
on A is an equivalence relation if it is
reflexive, symmetric and transitive. An example of such is equality on a
set. One might think of equivalence as a way to glob together elements that
can be considered the same relative to a property. That is elements become
indistinguishable relative to the relation. For example in arithmetic we
don't think twice about 1/2 and 2/4 as having the same value but they are
different objects. In geometry, similarity of
triangles is an equivalence relation.
A right angled triangle with legs of length 3 and 4 and hypotenuse
of length 5 is not the same as one with lengths 6, 8 and 10. Yet, we think
of them as equivalent because the ratios of the corresponding sides are the
same.
For each 
, we define the equivalency class containing a
to be the set of those elements which are equivalent to a. We denote this
set by [a] although you should be aware that there is no standard notation
for this. Some authors use 
, <a> and so on. The equivalence
classes have some interesting properties. First, no equivalence class is
empty since 
. Two equivalence classes are either
equal or disjoint. In fact, a and b are equivalent if and only if
[a] = [b]. Each element of A belongs to some equivalence class (
in fact ). The union of all the equivalence classes is A.
We say that the equivalence classes partition A.
A partition of A is a collection of non empty
disjoint subsets of
A and whose union is A. Thus the equivalence classes of a relation
are a partition.
Each equivalence relation on a set partitions the set into its equivalence
classes but also for each partition of the set there is an equivalence relation
whose equivalence classes are the sets in the partition. As an example, if
one partition is 
. Consequently, there is an
equivalence relation on A whose equivalence classes are these two subsets.
One problem is how should we refer to the equivalence classes? As we noted above [a] = [b] whenever a and b are equivalent. What we do is choose one and only one element of each equivalence class to represent the class. That element is called a class representative. The set of representatives is called the set of equivalence class representatives. There are usually many such ways to construct this set of representatives.
If is an equivalence relation on A then the factor set
denoted by 
is the set consisting of the equivalence classes of .
For the example we gave above, our equivalence classes are [1] and [3].
Thus the factor set is {[1], [3]}. We have chosen 1 to represent the
equivalence class {1,2} and (no choice) 3 to represent the equivalence class
{3}. Thus the set of class representatives is {1,3}. Note the similarity
with the factor set. The only difference is that the elements of the factor
set have [ and ] around the number (and of course are subsets of A whereas
the things in the set of class representatives are elements of A).
Because of this, we often abuse notation and write the set of class
representatives for the factor set, letting the context make it clear how we
are using a class representative.