Question

Let R be the relation over the set of all straight lines in a plane such that l₁ R l₂ ⇔ l ₁⊥ l₂. Then, R is
(a) symmetric
(b) reflexive
(c) transitive
(d) an equivalence relation

Answer 1

(a) symmetric

A = Set of all straight lines in the plane

$$\begin{gathered} R=\left\{\left(l_{1,}{l}_2\right):l_1,l_2\in A:l_1\perp l_2\right\} \\ \mathrm{Reflexivity}:l_1\mathrm{is}\mathrm{not}\perp l_1 \\ \Rightarrow \left(l_1,l_1\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Let}\left(l_1,l_2\right)\in R \\ \Rightarrow l_1\perp l_2 \\ \Rightarrow l_2\perp l_1 \\ \Rightarrow \left(l_2,l_1\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}\left(l_1,l_2\right)\in R,\left(l_2,l_3\right)\in R \\ \Rightarrow l_1\perp l_2\mathrm{and}{l}_2\perp l_3 \\ \mathrm{But}{l}_1\mathrm{is}\mathrm{not}\perp l_3 \\ \Rightarrow \left(l_1,l_3\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$