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Equivalence Class


Active member
Jan 8, 2013
Dear Everyone,

I am struck in writing the equivalence classes. And the problem is this:
Let ${\R}^{2}= \R \times \R$. Consider the relation $\sim$ on ${\R}^{2}$ that is given by $({x}_{1},{y}_{1}) \sim ({x}_{2},{y}_{2})$ whenever ${y}_{1}-{{x}_{1}}^{3}={y}_{2}-{{x}_{2}}^{3}$. Prove that $\sim$ is an equivalence relation. What are the equivalence classes?

I have proved that relation is an equivalence relation.

Here is my attempt:




Well-known member
MHB Math Scholar
Jan 16, 2013
Hi Cbarker1 ,

Since you have shown it is an equivalence relation, you know that every point in $\mathbb{R}^{2}$ must belong to an equivalence class. Consider points along the $x$-axis and see if you can determine the class to which they belong. Alternatively, you could also try points along the $y$-axis. In either case, this should help you figure things out.