# set theory – Transversal of an equivalence relation

Define a relation on $$mathbb{R}^2$$ by $$(a, b)sim(c, d)$$ if and
only if $$(c-a, d- b) in mathbb{Z}$$. Prove that $$sim$$ is an equivalence relation.
Identifying $$mathbb{R}^2$$ with the plane in the usual way, describe the most
natural transversal for $$sim$$ which you can find. What, if anything,
has this question to do with doughnuts?

I have already proven the equivalence relation, and know the solution of the second part, since it is provided, but I have problems with understanding it. I presume it’s asking about the equivalence classes?

Solution:

A natural transversal is $$I times I$$ where $$I = {r | r in mathbb{R}, 0 leq r < 1}$$. Join the top and bottom of $$I$$ to form a
tube, and bend the tube round to join the two circles as well. You
have the surface of a doughnut (a torus in mathematical language).