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.
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 line ‘L’ is parallel to itself, therefore (L, L) ∈ R
R is reflexive.
Symmetric : Let (L1, L2) ∈ R
L1 is parallel to L2
⇒ L2 is parallel to L1
⇒ (L2, L1) ∈ R
⇒ R is symmetric.
Transitive : Let (L1, L2) ∈ R and (L2, L3) ∈ R
⇒ L1 is parallel to L2 and L2 is parallel to L3
⇒ L1 is parallel to L3
⇒ (L1, L3) ∈ R
⇒ R is transitive.
Hence R is an equivalence relation.
Let B be the set of lines related to the line y = 2x + 4y
⇒ B = {L : L is a line parallel to y = 2x +4y}
∴ B = {L : L is a line whose equation is y = 2x + K, K being any real}