I was wondering if you could help me with this question?
"Consider the following relation on $\mathbb{R}$, the set of real numbers:
aRb $\iff_{def}$ a - b $\in \mathbb{Z}$
(a) Prove that this is an equivalence relation (done)
(b) Prove that the set $[\pi]_R$ = {x $\in \mathbb{R}$ | xR$\pi$} is in bijective correspondence with $\mathbb{Z}$"
I guess I have to find a function going from $\mathbb{Z}$ to $[\pi]_R$ and prove that it's bijective, but I have no idea of what this function would look like !
Thank you
What about $f:[\pi]_R \to \mathbb Z$ such that $x \mapsto x-\pi$?
Check that $f(x) \in \mathbb Z$ (it follows from the definition of $R$),
that $f$ is surjective (hint: given $k \in \mathbb Z$, what is $f(k+\pi)$?),
and that, for $x,y \in \mathbb R$ such that $xRy$, if $x-\pi = y -\pi$, then that $x=y$.