Let L be the set of all straight lines in the plane Define a

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.