In set notation, how can I express that: for an equivalence relation over $\mathbb{R}$ (say $a$R$b$ for example) one class includes non-horizontal and non-vertical lines passing through the origin, while all other classes consist of single lines.
For example, lines through the origin for arbitrary positive slope $c \gt 0$ could be expressed as:
$\forall a\in \mathbb{R} \ \ [a]=\{ \ b \in \mathbb{R} \ | \ b = c \cdot a \}$
But, how do I express that $c \ne \infty$ (i.e. that the line is not vertical), and what about the classes of a single line?
This is related to a problem from a proofs course for which I have determined the solution in English as above but have abstracted away the details of the problem seeing as I am merely unsure about how to formally express my solution and would ultimately like to apply the details myself and thus do the problem myself.
My professor said that: since the relation was of the form $x$R$y$ such that $x=a^n y$ for $n$ an integer, it was sufficient to denote the equivalence class as $[x] = \{y \ | \ a^n x = y\}$ since $a^{-n} x$ would be redundant for $n$ an integer.