Question

Mark the correct alternative in the following question:

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b for all a, b $\in$ T. Then, R is

a) reflexive but not symmetric (b) transitive but not symmetric
c) equivalence (d) none of these

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ R=\left\{\left(a,b\right):a\mathrm{is}\mathrm{congruent}\mathrm{to}b;a,b\in T\right\} \\ \mathrm{As},a\cong a \\ \Rightarrow \left(a,a\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{relation} \\ \mathrm{Let}\left(a,b\right)\in \mathrm{R}.\mathrm{Then}, \\ a\cong b \\ \Rightarrow b\cong a \\ \Rightarrow \left(b,a\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{relation} \\ \mathrm{Let}\left(a,b\right)\in R\mathrm{and}\left(b,c\right)\in R.\mathrm{Then}, \\ a\cong b\mathrm{and}b\cong c \\ \Rightarrow a\cong c \\ \Rightarrow \left(a,c\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{relation} \\ \therefore R\mathrm{is}\mathrm{an}\mathrm{equivalence}\mathrm{relation} \end{gathered}$$

Hence, the correct alternative is option (c).