Let $R$ be a relation on $\Bbb Q$ defined by \begin{equation*} aRb \iff a^3 + b^2 + a^2 b = b^3 + a^2 + ab^2, \end{equation*} for all $a,b \in \Bbb Q$.
I've shown that this relation $R$ is a equivalence relation. But, I got confused to determine the equivalence classes since it was on $\Bbb Q$. For the integers part I've understood, but not yet for the fractions part. Any ideas? Thanks in advanced.
We have \begin{equation*} aRb \iff a^3 + b^2 + a^2 b - (b^3 + a^2 + ab^2)=0\iff (a-b)(a+b)(a+b-1)=0 \end{equation*} So set of all numbers related to a given number $a$ is $\{a,-a,1-a\}$. So for example, $\frac12R(-\frac12)$, $(-\frac12)R\frac32$ but $\frac12\not R\frac32$.