Question

Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?

(a) Reflexive relation

(b) Transitive relation

(c) Symmetric relation

(d) Equivalence relation

I got this question in an internship interview.

My question comes from Types of Relations in chapter Relations and Functions of Mathematics – Class 12

Answer 1

Right answer is (d) Equivalence relation

For explanation: This is an equivalence relation. A relation R is said to be an equivalence relation when it is reflexive, transitive and symmetric.

Reflexive: We know that a line is always parallel to itself. This implies that I1 is parallel to I1 i.e. (I1, I2)∈R. Hence, it is a reflexive relation.

Symmetric: Now if a line I1 || I2 then the line I2 || I1. Therefore, (I1, I2)∈R implies that (I2, I1)∈R. Hence, it is a symmetric relation.

Transitive: If two lines (I1, I3) are parallel to a third line (I2) then they will be parallel to each other i.e. if (I1, I2) ∈R and (I2, I3) ∈R implies that (I1, I3) ∈R.