= Solution
For positive integer pairs, <reflexivity> is $ab=ba$. If $ad=bc$, reversing the equality gives $cb=da$, establishing <symmetry>. For <transitivity>, suppose $(a,b)R(c,d)$ and $(c,d)R(e,f)$. Then $ad=bc$ and $cf=de$, whence
$$
adf=bcf=bde.
$$
Since $d>0$, cancellation gives $af=be$. Thus $(a,b)R(e,f)$, and $R$ is an <equivalence relation>.
Membership in the <equivalence class> of $(4,6)$ means $4b=6a$, or $2b=3a$. Since $2$ and $3$ are <coprime>, $a=2r$ and $b=3r$ for a positive integer $r$. Conversely every such pair satisfies the equality. Therefore
$$
\boxed{[(4,6)]_R=\{(2r,3r):r\in\mathbb Z_+\}}.
$$
These pairs are precisely the positive numerator-denominator representations of the <rational number> $2/3$.
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