Practicing Success
Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. |
R is an equivalence relation R is transitive R is not an equivalence relation Cannot be determined |
R is an equivalence relation |
R= {$(T_1, T_2)$: $T_1$ is similar to $T_2$} R is reflexive since every triangle is similar to itself. Further, if $(T_1, T_2) ∈ R$, then T, is similar to $T_2$. ⇒ $T_2$ is similar to $T_2$ $⇒ (T_2, T_1) ∈ R$. Therefore, R is symmetric. Now, let $(T_1, T_2), (T_2, T_3) ∈ R$. ⇒ $T_1$ is similar to $T_2$ and $T_2$ is similar to $T_3 $ ⇒ $T_1$ is similar to $T_3$ $⇒ (T_1, T_3) ∈ R$ Therefore, R is transitive. Thus, R is an equivalence relation. |