So, let us start with the definition of a partition of a set. So, imagine you are given a set C which may be finite or it may be infinite. Now, what is the partition of this set C? The partition here is basically a collection of pairwise disjoint, non-empty subsets say m subsets of C which should be pairwise disjoint such that if you take their union, you should get back the original set C.

So intuitively, say for example, you have the map of India you can say that the various states of India partition the entire country India into various subsets such that there is no intersection among the states here. So, in that sense, I am just trying to find out some subsets of the set C such that there should not be any overlap among those subsets and if I take the union of all those subsets I should get back the original set C, there should not be any element of C which is missing.

More formally the requirements here are the following. Each subset ≠ ∅ that means each subset should have at least one element. They should be pairwise disjoint. That means if I take any two subsets, then their intersection must be empty, and their union should equal the set C. One trivial partition of the set C is the set C itself.

What we now want to establish here is a very interesting relationship between the equivalence from an equivalence relation to the partition of a set. So, imagine you are given a set C consisting of n elements. Now what I can prove here is that if R is an equivalence relation over the set C and if the equivalence classes which I can form with respect to the relation R are C1, ..., Cm. Then my claim here is that the equivalence classes C1, ..., Cm constitute a partition of the set C.

So, just to recall, the definition of partition demands me to prove three properties, the first property is that each of this subset should be non-empty. And that is trivial because I know that each of these equivalence classes is non-empty because each of these equivalence classes is bound to have at least one element, because my relation R is an equivalence relation, it will be a reflexive relation that means the element will be related to itself. That means none of these equivalence classes will be an empty set.

The second requirement from the partition is that the union of the various subsets should give me back the original set. So, my claim here is that if I take the union of all these m equivalence classes, I will definitely get back my original set C. And this is because you take any element x ∈ Ci, it is bound to be present in at least one equivalence class.

Third requirement from the partition was that the various subsets in the partition should be pairwise disjoint. So, in this specific case, I have to show that you take any two equivalence classes, they should be pairwise disjoint and that is easy because in the last lecture we proved that two equivalence classes are either same or they are disjoint.

Now, I can prove the property in the reverse direction as well. What do I mean by that? I claim here that you give me any partition of a set C, say you give me a collection of m subsets which constitute a partition of the set C. Then I can give you an equivalence relation R whose equivalence classes will be the subsets which you have given me in the partition.