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?

- ServerfaultXchanger
- SuperuserXchanger
- UbuntuXchanger
- WebappsXchanger
- WebmastersXchanger
- ProgrammersXchanger
- DbaXchanger
- DrupalXchanger
- WordpressXchanger
- MagentoXchanger
- JoomlaXchanger
- AndroidXchanger
- AppleXchanger
- GameXchanger
- GamingXchanger
- BlenderXchanger
- UxXchanger
- CookingXchanger
- PhotoXchanger
- StatsXchanger
- MathXchanger
- DiyXchanger
- GisXchanger
- TexXchanger
- MetaXchanger
- ElectronicsXchanger
- StackoverflowXchanger
- BitcoinXchanger
- EthereumXcanger