Suppose we have two oriented segments, (A, B) and (C, D) - apologies if the notation is weird, it is what my book uses. These two oriented segments are equivalent to each other. We can say then that the equivalent class of the segment (A, B) is equal to the equivalent class of (C, D), since they are both equivalent to each other.
Here is my question. When we state that, are we saying that there are two separated classes of equivalent segments, one for the segment (A, B) and one for the segment (C, D), and that they`re equal to each other, or are we saying that both (A, B) and (C, D) belong to the same class of equivalent oriented segments.
I am asking this because my professor, when explaining this, represented a class for (A, B) and a class for (C, D), which led me to believe that there are two separated classes for each segment.
