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

Answer: (C)

equivalence

Let there be three triangles a,b,c in Euclidean plane such that aRb and bRc.

We know that if triangle a is congruent to triangle b and triangle b is congruent to triangle c, triangle a must also be congruent to triangle c .

So, aRc and thus relation R is transitive .

Also we know that any triangle (say t) in Euclidean plane is always congruent to itself .

So, tRt and thus relation R is reflexive .

Let there be two triangles n and m in Euclidean plane.

We know that if n is congruent to m, m must also be congruent to n.

So, $nRm\iff mRn$ and thus relation R is symmetric .

Thus, finally we arrive at conclusion that realation R is equivalence relation .