Question

Let S be the set of all straight lines in a plane. A relation R is defined on S by $aRb\iff a\perp b$ then R is :

• A. reflexive but neither symmetric nor transitive
• B. symmetric but neither reflexive nor transitive
• C. transitive but neither reflexive nor symmetric
• D. an equivalence relation

Answer 1

Answer: (B)

We have to test the equivalencity of relation R on S.
(1) Reflexivity :
In a plane any line be parallel to itself not perpendicular. Hence aRb , R is not reflexive.
(2) Symmetry :
In a plane if a line AB is perpendicular to the other line BC, then BC is also perpendicular to AB, i.e.,
$aRb\Rightarrow AB\perp BC$
And $bRa\Rightarrow BC\perp AB$
Hence R is symmetric.
(3) Transitivity :
In a plane, let AB, BC and CA be three lines, such that
$AB\perp BC$ and $BC\perp CD$
$\Rightarrow AB||CD\Rightarrow aRb$, R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.