Equivalence Relations (again)

If I may, I'd like to just pop in here and take another stab at explaining equivalence relations from the point of view of partitioning a set.

Let's take a set with elements:

Assume that each of these dots represents a distinct element in the set. We know, because we are familiar with the concept of equality, that each of these elements is equal to itself and only to itself. We can show this on the picture of the set by drawing boundaries around each element to separate it from all of the others:

Now, we can say, in terms of this picture, that two elements are equal if they are contained within the same boundary. This is the same as saying that each element is equal only to itself.

From here we can generalize the concept of equality by allowing more than one element inside the boundaries. So we could do something like this:

These boundaries no longer show equality, but some generalization of it that we call equivalence. In terms of the picture, two elements are equivalent if they are within the same boundaries. These boundaries can be whatever we need them to be. If the set is the set of integers, we could put all of the evens in one boundary and all of the odds in another. Under that example, two elements would be equivalent if they are both even or both odd.

Each section of elements is what we call an equivalence class. Under the odd/even example, there are two equivalence classes - one set with all of the odds and another with all of the evens. Under equality, there are as many equivalence classes as there are elements in the set - one for each element.

Earlier, you already showed that this idea of equivalence is actually a special type of relation between elements that is reflexive, symmetric, and transitive. Subsequently, I showed that any relation that is reflexive, symmetric, and transitive as defined on a set results in these kinds of internal boundaries on the set.

I hope that this helps somewhat.