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 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.