Let $S^1 = \{ \omega \in \mathbb{C}: \ |\omega| =1 \}$. I need to figure out the relation $\sim$ between two elements of a set $S^1 \times \mathbb{R}$ such that quotient group $S^1 \times \mathbb{R} / _{\sim}$ will be izomorphic with torus.
Please help. Thanks Tommy.
You've already deduced a scheme for equivalence relations that will do the job, but I would suggest a few refinements.
First of all, there's no need for the absolute value bars--we can simply say $u-t=k\cdot l$, instead, and it will make it easier to prove transitivity, without making symmetry particularly difficult to prove.
Second of all, there's no need to leave $l$ general, nor even think of it as a length. Any non-zero real number will do, once we've dropped the absolute value bars, but I'd pick $l=1$ for simplicity. That reduces the condition to $u-t=k.$