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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 ∀ a, b ∈ T. Then R is ______.
Options
Reflexive but not transitive
Transitive but not symmetric
Equivalence
None of these
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Solution
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 ∀ a, b ∈ T. Then R is equivalence.
Explanation:
Given aRb, if a is congruent to b, ∀ a, b ∈ T.
Then, we have aRa ⇒ a is congruent to a; which is always true.
So, R is reflexive.
Let aRb ⇒ a ~ b
b ~ a
bRa
So, R is symmetric.
Let aRb and bRc
a ~ b and b ~ c
a ~ c
aRc
So, R is transitive.
Therefore, R is equivalence relation.
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