# Equivalence relation for which there are infinitely many equivalence classes.

On set $\mathbb{R}$ and the relation on it where $x\sim y$ if $x^{4}=y^{4}$. Then $\sim$ is equivalence relation for which there are infinitely many equivalence classes, one of which consists of a single element and, and the rest of two elements. How to go about evaluating equivalence classes of $\sim$

• What do you mean by "evaluate" here? – Travis May 25 '15 at 8:44
• finding the equivalence classes – Roneel Kumar May 25 '15 at 8:48
• If $x \sim y$, then $x^4 = y^4$. How can you solve for, say, $x$? – Travis May 25 '15 at 8:52
• For instance, the real fourth roots of $\frac35$ will forn one equivalence class, and the real fourth roots of $0$ will form another equivalence class; however, the real fourth roots of $-1$ will not form an equivalence class. – bof May 25 '15 at 8:54

## 1 Answer

The quotient set is $$\mathbb{R}/\sim~~=\{\{a,-a\}|a \in \mathbb{R}\}$$

because $x \sim y$ if and only if $x=\pm y$