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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,bT. Then, R is

A
reflexive but not symmetric
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B
transitive but not symmetric
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C
equivalence
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D
none of these
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Solution

The correct option is B 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, nRmmRn and thus relation R is symmetric .

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

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