Question

Let L be the set of all lines in a plane and R be the relation in L defined as

R = {(L1, L2) : L1is perpendicular to L2}. Show that R issymmetric but neitherreflexive nor transitive​

Answer 1

Solution: R is not reflexive, as a line L1 can not be perpendicular to itself, i.e., (L1, L1)∉ R. R is symmetric as (L1, L2) ∈ R⇒ L1 is perpendicular to L2⇒ L2 is perpendicular to L1⇒ (L2, L1) ∈ R.

R is not transitive. Indeed, if L1 is perpendicular to L2 andL2 is perpendicular to L3, then L1 can never be perpendicular toL3

In fact, L1 is parallel to L3i.e., (L1, L2) ∈ R, (L2, L3) ∈ R but (L1, L3) ∉ R.