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.

We have, L is the set of lines.

R = {(L₁, L₂) : L₁ is parallel to L₂} be a relation on L

Now,

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

Since a line is always parallel to itself.

∴ (L₁, L₂) ∈ R

⇒ R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let L₁, L₂∈ L and (L₁, L₂) ∈ R

⇒ L₁ is parallel to L₂

⇒ L₂ is parallel to L₁

⇒ (L₁, L₂) ∈ R

⇒ R is symmetric

Transitive: For Transitivity, we need to prove that-

If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

Let L₁, L₂ and L₃∈ L such that (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

Since, R is reflexive, symmetric and transitive, so R is an equivalence relation.

And, the set of lines parallel to the line y = 2x + 4 is y = 2x + c For all c ∈ R

where R is the set of real numbers.