Partition induced by the Equivalence Relation

Question

I'm not sure I understand this concept.

Let's say we have a "Is parallel to" relation from the set of all lines in the Cartesian plane. What would be the partition induced by this relation?

Thank you.

Answer 1

Two lines in the plane are parallel to each other iff they have the same slope. So a partition representative can be given as a Real number which is the slope of the collection of parallel lines with the same slope.

EDIT: as Will Jagy wrote, you also need to consider the lines with infinite slope, you can choose , say, a representative $x=k$ , where $k$ is constant.

Answer 2

The partition related to the relation $\sim$ on a set $S$ is given by the set of subsets of $S$ of the form $S_x = \{y \in S \mid x \sim y\}.$

Here, $S$ is the set of all lines in the plane, and the subsets forming the partition have the form $S_l = \{l' \in S \mid \text{l is parallel to l'} \}.$ Each such subset is in a one-to-one correspondence with the slope of any and each of the lines belonging to it. In other words, you have a bijection $(S/\sim) \cong \mathbb{\hat{R}} = \mathbb{R} \cup \{\infty\},$ under which $S_l$ corresponds to the slope of $l.$

Answer 3

Not entirely clear what you want, but a nice set of representatives is pairs of antipodal points on the standard unit circle $x^2 + y^2 = 1.$ For any family of parallel lines, there is one line that passes through the origin; it also passes through two points on the unit circle that are opposite ends of a diameter.

It does not suffice to use the concept of slope, vertical lines have infinite or undefined slope.