Equivalence classes for a Relation on a product set.

Question

How do you determine the the equivalence classes for a relation on a product set?

Background:

Let $S=\left\{1,2,3,4\right\}$ and $A=S\times S$. The relation $R$ on $A$ can be defined by

$$\left(a,b\right)R\left(c,d\right) \iff a/b =c/d$$

For example:

$$\left(1,2\right)R\left(2,4\right) \text{ since } 1/2 = 2/4$$

Assuming $R$ is an equivalence relation, what are the equivalence classes for $A/R$?

Answer 1

Just do it one by one: (I use $\sim$ for $R$)

$(1, 1) \sim (2, 2) \sim (3, 3) \sim (4, 4)$

$(1, 2) \sim (2, 4)$

$(1, 3)$

$(1, 4)$

$(2, 1) \sim (4, 2)$

$(2, 3)$

$(3, 1)$

$(3, 2)$

$(3, 4)$

$(4, 1)$

$(4, 3)$