question_answer 14)

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4. 

Answer:

Let A be the set of all lines in xy plane       R = {L1 , L2} : L1 is parallel to L2}       R = {(L1, L2} : L1 || L2}       Reflexive : Since every time ?L? is parallel to itself, therefore (L, L)        is reflexive.       Symmetric : Let (L1, L2)        is parallel to L2        is parallel to L1             Transitive : Let (L1, L2)  and (L2, L3)       is parallel to L2 and L2 is parallel to L3        is parallel to L3              is transitive.       Hence R is an equilivalence relation.       Let B be the set of lines related to the line y = 2x + 4.       is a line parallel to y = 2x + 4}       whose equation is       y = 2x + K, K being any real}.