List the elements in the equivalence class $E_{(9,2)}$

Question

Let $A = \mathbb{N} \times \mathbb{N}$ and define a relation $R$ on $A$ by $(a, b) R (c, d )$ iff $ab = cd$.

Obviously $R$ is an equivalence relation.

Problem: List the elements in the equivalence class $E_{(9,2)}$.

Modified

$E_{(9,2)} = \{(9,2),(2,9),(1,18),(18,1),(6,3),(3,6)\}$.

Is this correct?

Answer 1

It is not correct. First, elements of the equivalence class of $(9,2)$ are, like $(9,2)$ itself, elements of $A$, not elements of $A\times A$: they are simply ordered pairs of natural numbers. $(9,2)$ and $(2,9)$ are elements of that class; the ordered pairs of ordered pairs that you wrote down are not.

Secondly, you missed four elements: $(1,18),(18,1),(3,6)$, and $(6,3)$. Each of these pairs is in the same class as $(9,2)$, because in each pair the elements have the same product as those of $(9,2)$, namely, $18$.

Answer 2

No. The pairs $(3, 6)$ and $(6, 3)$ are in the same equivalence class. Can you see why? There are also two other pairs in there...try to find them.