Recall the relation on the set
One of the equivalence classes is which is equivalent to writing
We could do this because the equivalence class collects all the natural numbers that are related to zero under the relation
The following theorem generalises this idea for any relation on the set for the integer
Let be an equivalence relation on set If then
Essentially, equivalence classes are equal if the elements are related under the relation And simultaneously, knowing that elements are related under means their equivalence classes are equal.
An equivalence class divides set a into equivalence classes. We call this situation a partition of set
A partition of a set is defined as a set of non-empty subsets of such that both these conditions are simultaneously satisfied:
(i) the union of all these subsets equals
(ii) the intersection of any two different subsets is
Let’s return to our example: on the set We could represent this set as:
- NOTE: Each equivalence class above represents an infinite set and despite the drawing suggesting is larger than for instance, this is not true.