Let L be the set of all lines in XY plane and R be the relation in L d

1.	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» (L₁, L₁) ∈ R ⇒ L₁ \|\| L₁ which is true (L₁, L₁) ∈ R, hence reflexive (L₁, L₂) ∈ R ⇒ L₁ \|\|L₂ ⇒ L₂ \|\|L₁ ⇒ (L₂, L₁) ∈ R, hence symmetric L₁, L₂) ∈ R and (L₂, L₃) ∈ R ⇒ l₁ \|\| L₂, L₂ \|\|L₃ ⇒ L₁ \|\| L₃ ⇒ (L₁, L₃) ∈ R hence R is transitive hence it is equivalence relation Let L be the required line = {L : L is parallel to y = 2x + 4} = {L:L is a line whose relation is y : 2x + k where k is any real}

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»

(L₁, L₁) ∈ R ⇒ L₁ || L₁ which is true

(L₁, L₁) ∈ R, hence reflexive

(L₁, L₂) ∈ R ⇒ L₁ ||L₂

⇒ L₂ ||L₁

⇒ (L₂, L₁) ∈ R, hence symmetric

L₁, L₂) ∈ R and (L₂, L₃) ∈ R

⇒ l₁ || L₂, L₂ ||L₃

⇒ L₁ || L₃ ⇒ (L₁, L₃) ∈ R

hence R is transitive hence it is equivalence relation

Let L be the required line

= {L : L is parallel to y = 2x + 4}

= {L:L is a line whose relation is y : 2x + k where k is any real}