For the set $ \Bbb R$, define two elements in $ \Bbb R$ to be equivalent if their difference belongs to $ \Bbb Q$.
I can prove that this defines an equivalence relation. (see Verify an equivalence relation.)
However, I would like to be able to define the class and the partition. Is it possible to place an element into a class without comparing it to other members of the class?
A similar problem, where the difference belongs to $ \Bbb Z$, any element r $\in \Bbb R$ can be classified by setting up class Am such that m = r - g(r) where g(r) returns the nearest integer less than r. As a result, any r $\in \Bbb R$ can be put into some class Am, $0<=m<1$. There may be other indexing schemes.
But I can't come up with anything for the difference belonging to $ \Bbb Q$.
This problem is from Pinter's A Book of Abstract Algebra.