## 1.6 Partitions

**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

partitionof 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.