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 = {(L₁, L₂) : L₁ is parallel to L₂}

R is reflexive as any line If (L₁, L₂) ∈ R, then

⇒ L₁ is parallel to L₂.

L₁ is parallel to itself i.e., (L₁, L₂) ∈ R

⇒ L₂ is parallel to L₁.

⇒ (L₂, L₁) ∈ R

∴ R is symmetric.

(L₁, L₂), (L₂, L₃) ∈ R

⇒ L₁ is parallel to L₂

⇒ L₂ is parallel to L₃

∴ L₁ is parallel to L₃.e

⇒ (L₁, L₃) ∈ 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

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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