Question

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 1

$\text{\bf{equivalence relation}}$:- A relation R is said to be an equivalence relation. if it is simultaneously reflexive, symmetric and transitive on A.

It is given that the relation in L defined as
R = {(L₁, L₂): L₁ is parallel to L₂}
R is reflexive as any line L₁ is parallel to itself
⇒ (L₁, L₂) ∈ R
Now, Let (L₁, L₂) ∈ R
⇒ L₁ is parallel to L₂.
⇒ L₂ is parallel to L₁.
⇒ (L₂, L₁) ∈ R
Therefore, R is symmetric.
Now, Let (L₁, L₂), (L₂, L₃) ∈ R
⇒ L₁ is parallel to L₂. Also, L₂ is parallel to L₃.
⇒ L₁ is parallel to L₃.
⇒ (L₁, L₃) ∈ R
Therefore, R is transitive.
Therefore, R is an equivalence relation.

The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line
y = 2x + 4
Slope of line y = 2x + 4 is m = 2
We know that slopes of parallel lines are same.
The line parallel to the given line is of the form y = 2x + c where, c ∈ R.
Therefore, the set of all lines related to the given line by y = 2x + c, where c ∈ R.