Equivalence Classes: Solving Cbarker1's Problem

[cbarker1]

Dear Everyone,

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

 

[GJA]

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.