Equivalence relation on $\mathbb{R}$: $x\sim y \iff x-y\in\mathbb{Q}$ - disjoint equivalence classes

by rbird   Last Updated August 04, 2018 04:20 AM

Consider the equivalence relation on $\mathbb{R}$ given by $x\sim y \iff x-y\in\mathbb{Q}$.

Several questions about this have been asked before, but I could not find anywhere what the disjoint equivalence classes of $\mathbb{R}/\sim$ are.

I know that the equivalence classes are of the form $$[a]=\{a+q:q\in\mathbb{Q}\}$$ where $a$ is irrational.

But how would I go about writing down $\mathbb{R}$ as a union of disjoint equivalence classes? (I believe this is possible for any equivalence relation on any set).

I want to write $$\mathbb{R}=\mathbb{Q}\sqcup \bigsqcup_{\text{some condition on $a$}}[a]\,.$$

I know that the condition cannot be $a\notin\mathbb{Q}$, because then, for example, $[\pi+1]=[\pi]$, so the union is not disjoint.

What condition on $a$ makes this work? Is it $\{a\in[0,1):a\notin\mathbb{Q}\}$? If so, can anyone give a hint for how to prove that these are all disjoint?



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