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 $a R b$ 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
We have,
$R=\{(a, b): a$ is congruent to $b ; a, b \in T\}$
As, $a \cong a$
$\Rightarrow(a, a) \in R$
So, $R$ is reflexive relation
Let $(a, b) \in \mathrm{R}$. Then,
$a \cong b$
$\Rightarrow b \cong a$
$\Rightarrow(b, a) \in R$
So, $R$ is symmetric relation
Let $(a, b) \in R$ and $(b, c) \in R$. Then,
$a \cong b$ and $b \cong c$
$\Rightarrow a \cong c$
$\Rightarrow(a, c) \in R$
So, $R$ is transitive relation
$\therefore R$ is an equivalence relation
Hence, the correct alternative is option (c).
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