- #1

cbarker1

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MHB

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$\newcommand{\R}{\mathbb{R}}$

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:

$(a,b)=\left\{(x,y)\in{\R}^{2}|y-{x}^{3}=b-{a}^{3}\right\}$

Thanks

Cbarker1