Discrete Math - Equivalence Classes of a set containing all real numbers

Question

I'm trying to understand a problem given to me that asks the following:

Let A=R. For all x,y belonging to A, define x R y if and only if |x|=|y|. Determine if R is an equivalence relation. If yes, find all distinct equivalence classes

I have already proven that the relation is in fact an equivalence relation, but i am having trouble defining the distinct equivalence classes.

Would it be : {-infinity, infinity} or {x,-x} for each number within the set?

or am i completely wrong?

Answer 1

You're mostly right, except $\infty$ isn't a real number (and neither is $-\infty$): the equivalence classes are exactly the sets of the form $$\{x, -x\}$$ for $x\in\mathbb{R}$.

Note that not every equivalence class contains two elements: the equivalence class of $0$ is $\{0, -0\}=\{0\}$.