Supply an equivalence relation on R whose equivalence classes are $\{[m,m+1)|m\in Z\}$

Question

Supply an equivalence relation on R whose equivalence classes are $\{[m,m+1)|m\in Z\}$

Define $\sim :R\to R$ as

$\forall x,y\in R \\ x \sim y \iff \exists m\in Z \ni x,y \in [m,m+1)$

Will this suffice?

Answer 1

Clearly we have no problem with reflection and simmetry.

Transitivity:

$x\sim y$ and $y\sim z$ then $x,y\in [m,m+1)$ (and thus $[y]=m$) and $y,z\in [n,n+1)$ (and thus $[y]=n$) so $m=n$ and thus $x,z\in [m,m+1)$ so $x\sim z$.

Answer 2

Your definition is good, but it’s tautological.

Hint: this is linked to the greatest integer function. Define $\lfloor x\rfloor$ to be the largest integer less than or equal to $x$.

Next, $x\sim y$ is $\lfloor x\rfloor= \lfloor y\rfloor$.