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={(ℓ1,ℓ2):ℓ1 is parallel to ℓ2}. On the basis of the above information, answer the following: (i) Find whether the relation R is symmetric or not. (ii) Find whether the relation R is transitive or not. (iii) 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: (i) symmetric; (ii) transitive; (iii) {y=3x+c:c∈R}
Solution
(i) Symmetric: if ℓ1 is parallel to ℓ2, then ℓ2 is parallel to ℓ1; so (ℓ1,ℓ2)∈R⇒(ℓ2,ℓ1)∈R. Hence R is symmetric. (ii) Transitive: if ℓ1∥ℓ2 and ℓ2∥ℓ3, then ℓ1∥ℓ3; so (ℓ1,ℓ2),(ℓ2,ℓ3)∈R⇒(ℓ1,ℓ3)∈R. Hence R is transitive. (iii) Lines parallel to y=3x+2 have the same slope 3 but a different intercept; the whole family of related lines is
