Define a relation $\sim$ on $\mathbb{R}$ by the rule $x \sim y$ if $x - y \in \mathbb{Q}$. List 5 elements of the equivalence class $[\pi]$

Question

Part(a):List 5 elements of the equivalence class $[\pi]$

I have that $[\pi]=\{x\in \mathbb{R}\mid xR\pi\}=\{x\in \mathbb{R}\mid x-\pi\in\mathbb{Q}\}=\{\pi\}$.

Not sure this is correct because there are not 5 elements listed. Also is it correct for it to be written $x-\pi \in\mathbb{Q}$?

Part(b): Determine the equivalence class $[ 1 ]$.

I have that $[1]=\{x\in \mathbb{R}\mid xR1\}=\{x\in \mathbb{R} \mid x-1\in \mathbb{Q}\}=\{x\in \mathbb{R}\mid x \in \mathbb{Q}\}=\mathbb{Q}$.

Answer 1

Notice that for any $r \in \mathbb{Q}$, $\pi + r \in [\pi]$ since $\pi + r - \pi = r \in \mathbb{Q}$.