# 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

**Solution:**

R = {(L_{1}, L_{2}) : L_{1 }is parallel to L_{2}}

R is reflexive as any line If (L_{1}, L_{2}) ∈ R, then

⇒ L_{1} is parallel to L_{2}.

L_{1} is parallel to itself i.e., (L_{1}, L_{2}) ∈ R

⇒ L_{2} is parallel to L_{1}.

⇒ (L_{2}, L_{1}) ∈ R

∴ R is symmetric.

(L_{1}, L_{2}), (L_{2}, L_{3}) ∈ R

⇒ L_{1} is parallel to L_{2}

⇒ L_{2} is parallel to L_{3}

∴ L_{1} is parallel to L_{3}.e

⇒ (L_{1}, L_{3}) ∈ R

∴ R is transitive.

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 the line y = 2x + 4 is m = 2.

Line parallel to the given line is in the form y = 2x + c, where c ∈ R.

The set of all lines related to the given line is given by y = 2x + c, where c ∈ R

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 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

**Summary:**

Hence it is proved that 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