Finding the equivalence class of an equivalence relation

Question

I am not quite sure what this question means.

Are these equivalence classes valid as each element within a set can be rounded up to become the same integer?

{0.5,0.51}, {1.5, 1.51}, {2.5,2.51}, {3.5,3.51}, {4.5,4.51}

Answer 1

Some hints:

Given an equivalence relation it partitions the given ground set $X$ (the interval $[0,5]$ in the present example) into disjoint subsets. These subsets are called equivalence classes. Two elements $x$, $y\in X$ belong to the same class iff they are equivalent. In the given example the numbers $\pi$ and ${1\over\pi}$ are not equivalent (check this!), hence they belong to different equivalence classes. The equivalence class containing the number $4$ contains all real numbers $x$ satisfying $3<x\leq4$.

Answer 2

Hint:

If $R$ is a relation on some set $X$ and is prescribed by:$$xRy\iff f(x)=f(y)$$where $f$ denotes some function that has $X$ as codomain then $R$ is an equivalence relation and the equivalence class represented by $x$ is the set: $$f^{-1}(\{f(x)\})=\{y\in X\mid f(y)=f(x)\}$$