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 $b\forall a,b\in T$. Then $R$ is

• A. reflexive but not transitive
• B. transitive but not symmetric
• C. equivalence
• D. None of these

Answer 1

Answer: (C)

$(i)$ We know that every triangle is congruent to itself.
$\therefore (T_1$, $T_1)\in R$ for all $T_1\in T$. Thus, $R$ is reflexive.
$(ii)$ Let $(T_1$, $T_2)\in R$
$\Rightarrow T_1$ is congruent to $T_2$.
$\Rightarrow T_2$ is congruent to $T_1$.
$\therefore (T_2$, $T_1)\in R$
Thus, $R$ is symmetric.
$(iii)$ Let $(T_1$, $T_2)\in R$ and $(T_2$, $T_3)\in R$.
$\Rightarrow T_1$ is congruent to $T_2$ and $T_2$ is congruent to $T_3$.
$\therefore T_1$ is congruent to $T_3$
$\Rightarrow (T_1$, $T_3)\in R$.
Thus, $R$ is transitive.
$\therefore R$ is an equivalence relation.