# If $X = \{1, 2, 3, 4\}$ show there are just two equivalence relations on $X$ with $1\sim 2$ and $2 \sim3$

If $X = \{1, 2, 3, 4\}$ and $\sim$ is an equivalence relation on $X$, then if $1 \sim 2$ and $2 \sim 3$ show that there are just two possibilities for the relation $\sim$ and describe both relations.

This was a bonus question assigned on our last test in proofs. We were studying sets and equivalence relations in particular. As it is a proofs class, there must be proof statements and such included. I wasn't able to even begin to figure this question out though.

• Hint: an equivalence relation can be defined by specifying the equivalence classes. – Mark Bennet Mar 2 '14 at 20:28

• I see.. so .. S={1,2,3} and S"={4} or S={1,2,3,4} but in a formal proof how would go about actually SHOWING this, I mean a partitioned diagram is one thing but a full proof is another don't you think? – user122661 Mar 4 '14 at 17:38