MHB Equivalence Classes: Solving Cbarker1's Problem

AI Thread Summary
The discussion focuses on proving that the relation defined on R² by the condition (y1 - x1³ = y2 - x2³) is an equivalence relation. Cbarker1 has successfully demonstrated this and seeks to identify the equivalence classes. The suggested approach involves analyzing points along the x-axis or y-axis to determine their respective classes. This method aims to clarify the structure of the equivalence classes formed by the relation. Understanding these classes is essential for further exploration of the problem.
cbarker1
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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
 
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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.
 
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