Counting ordered pairs in quotient set

Question

We have equivalence relation E on the set $$A = \{1,2,3,4,5\}$$ So the quotient set: $$A/E = \{\{1,2,3\}, \{4\}, \{5\}\}.$$

How much orderd pairs we can find in E?

How to count the ordered pairs?

Thank you!

Answer 1

You should first prove, to yourself, the following general fact.

If $E$ is an equivalence relation on a set $X$, and $P$ is the partition induced by $E$ (the quotient set, as you call it), then for every $S\in P$, $S\times S\subseteq E$. Moreover, $E=\bigcup_{S\in P}S\times S$.

Now the solution is easy.