Question

Students of a school are taken to a railway museum to learn about railways heritage and its history. 

An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by

R = `{(l_1, l_2) : l_1  "is parallel to"  l_2}` 

On the basis of the above information, answer the following questions:

1. Find whether the relation R is symmetric or not.
2. Find whether the relation R is transitive or not.
3. If one of the rail lines on the railway track is represented by the equation y = 3x + 2, then find the set of rail lines in R related to it. 

Answer 1

1. Given Relation, R = `{(l_1, l_2) : l_1  "is parallel to"  l_2}` 
   ∵ l₁ || l₂ ⇒ l₂ || l₁
   Hence, given relation is symmetric. 
2. Let l₁ || l₂ and l₂ || l₃ ⇒ l₁ || l₃
   [As parallel lines are parallel to each other]
3. Given equation of line : y = 3x + 2
   Comparing with y = mx + C
   m = slope = 3
   Slope of parallel, lines is same
   ∴ Equation of set of rall line is given by 
   y = 3x + λ       λ ∈ R