What are the properties of this relation?

Question

If a relation $j R k$ is defined iff $\{p: p$ is prime and $p \mid j\} = \{q: q$ is prime and $q\mid k\}$, what are the properties of this relation?

It is definitely reflexive, symmetric. But I was wondering is this anti-symmetric and transitive?

Answer 1

Hint. As regards anti-symmetric property, is $25\ R\ 5$?

Note that the set $P_n:=\{p: p \mbox{ is prime and } p|n\}$ is made of the primes which appear in the factorization of $n$.

Answer 2

This is definitely symmetric and reflexive.

To see that this is transitive, let $a\ R\ b$ and $b\ R\ c$. Then, $\{p : p \text{ prime}, p | a\} = \{q : q \text{ prime}, q | b\}$, and $\{q : q \text{ prime}, q | b\} = \{r : r \text{ prime}, r | c\}$. It is easy to see here that $\{p : p \text{ prime}, p | a\} = \{r : r \text{ prime}, r | c\}$, hence $a\ R\ c$.

Hence the relation is an equivalence relation. The equivalence classes under this relation, is indexed by sets of primes, $E_{\{p_1,p_2,\ldots,p_n\}} = \{ p_1^{k_1}\ldots p_n^{k_n} : k_i \geq 1 \ \forall i\}$.