Chapter 8.3, Problem 31ES

In 19—31, (1) prove that the relation is an equivalence relation, and (2) describe the distinct equivalence classes of each relation.

31. Let P be the set of all points in the Cartesian plane except the origin. R is the relation defined on P as follows: For every $p_1$ and $p_2$ in P, $p_{1}R{p}_2\iff p_1$ and $p_2$ lie on the same half-tine emanating from the origin.