My question is:
How many different equivalence relations can we define on the set $A = \{x,y,z\}$?
I know that an equivalence relation is a relation that is symmetric, reflexive, and transitive, so how many I go about considering these possible relations?
I am particularly curious about this relation: $\{(x,x), (y,y), (z,z)\}$ Is it one possibility?
An equivalence relation is defined by its equivalence classes. Given an equivalence relation, its equivalence classes constitute a partition of the set $A$.
Hence, an easy way to count the number of equivalence relations is to count the number of ways in which $A$ can be partitioned. This is provided by the Bell number $B_3 = 5$.
An equivalence relation separates the set into equivalence classes, so you are looking at the number of ways to separate $\{x,y,z\}$ into groups. Your example is the identity relation, where each element is only related to itself. Yes, it is an equivalence relation. It separates the set into three equivalence classes, one for each element.
This task is solved here. The elements of the example you mentioned must be part of every equivalence relation, due to reflexivity. If you are not too familiar with this concept, write down all the reflexive pairs, than add some symmetric ones and see how far you can go.
