Let L be the set of all straight lines in the plane. Define a relation P on L as follows For all l_i, l_2 epsilon L, l_1 Pl_2 iff l_1 is parallel to l_2. Prove that P is an equivalence relation on L and describe the equivalence classes of P. Solution 1. For any line, l lPl ie l is parallel to itself Hence, P is reflexive 2. For any two lines: l,r lPr means rPl ie l parallel to r means r parallel to l Hence, P is symmetric 3. Let, l1 P l2, l2 P l3 Hence, l1 is parallel to l2 and l2 is parallel to l3 Hence, l1 is parallel to l3 Hence,l1 P l3 Hence, R is transitive. Consider the line: y=mx . All lines of slope m will be parallel to this So equivalence classes are :y=mx . For each different real number, m we have a different equivalence class..