TITLE: Finding the equivalence classes of a relation R
QUESTION [7 upvotes]: Let $A = \{0,1,2,3,4\}$ and define a relation $R$ on $A$ as follows:
$$R = \{(0,0),(0,4),(1,1),(1,3),(2,2),(3,1),(3,3),(4,0),(4,4)\}.$$
Find the distinct equivalence classes of $R$.

How do I solve this problem? What is an equivalence class? As I understand it so far, the equivalence class of $a$, is the set of all elements $x$ in $A$ such that $x$ is related to $a$ by $R$. What does this mean in my problems case?

REPLY [2 votes]: Let ={0,1,2,3,4} and define a relation  on  as follows:
={(0,0),(0,4),(1,1),(1,3),(2,2),(3,1),(3,3),(4,0),(4,4)}.
Find the distinct equivalence classes of .
These are actually really fun to do once you get the hang of them! 
The way I think of equivalence classes given a set of ordered pairs as well as given a set A, is what is related to what. First, I start with 0, and ask myself, which ordered pairs in the set R are related to 0?
Here it goes! (think of equivalence class as x in an ordered pair y, and the equivalence class of x is what x is related to in the y value of the ordered pair)
[0]: 0 is related 0 and 0 is also related to 4, so the equivalence class of 0 is {0,4}.
[2]: 2 is related to 2, so the equivalence class of 2 is simply {2}.
[3]: 3 is related to 1, and 3 is also related to 3, so the equivalence class of 3 is {1,3}.
[4]: 4 is related to 0, and 4 is also related to 4, so the equivalence class of 4 is {0,4}. 
Notice that the equivalence class of 0 and 4 are the same, so we can say that [0]=[4], which says that there are only three equivalence classes on the relation R.