Mathematics - Sets and Relations Question with Solution | TestHub

Let $S$ denote the set of all triangles in a plane. Let $R$ be the relation on $S$ defined by $\left({\Delta }_1,{\Delta }_2\right)\in$ $R$ $\Rightarrow$ triangle ${\Delta }_1\cong {\Delta }_2$ : (i) Let any triangle $\Delta \in S$, we have ${\Delta }_1\cong {\Delta }_2\Rightarrow (\Delta ,\Delta )\in R\forall \Delta \in S\Rightarrow R$ is reflexive on (ii) Let ${\Delta }_1,{\Delta }_2\in \mathrm{S}$, such that $\left({\Delta }_1,{\Delta }_2\right)\in R$, then ${\Delta }_1\cong {\Delta }_2\Rightarrow {\Delta }_2\cong {\Delta }_1\Rightarrow \left({\Delta }_2,{\Delta }_1\right)\in R\Rightarrow R$ is symmętric (iii) Again, let ${\Delta }_1,{\Delta }_2,{\Delta }_3\in \mathrm{S}$ such that $\left({\Delta }_1,{\Delta }_2\right)$ $\in R$ and $\left({\Delta }_2,{\Delta }_3\right)\in R\therefore {\Delta }_1\cong {\Delta }_2\cong {\Delta }_3$ $\therefore \left({\Delta }_1,{\Delta }_2\right)\in R$ $\Rightarrow R$ is transitive. $\therefore \mathrm{R}$ is an equivalence relation.