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 ϵ T. Then, R is

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

Answer 1

C. equivalence

R: a R b ⟺ a ≅ b

Since, every triangle a∈T is congruent to itself, therefore (a, a)∈R ∀a∈T. Hence, R is reflexive.

If a ≅ b, then b ≅ a. Hence if (a, b)∈R, then (b, a)∈R ∀a, b∈T. Hence, R is symmetric.

If a ≅ b and b ≅ c, then a ≅ c. Hence if (a, b) and (b, c) belongs to R, then (a, c) will belong to R ∀a, b, c∈T. Hence, R is transitive.

Since R is reflexive, symmetric and transitive, therefore R is equivalence relation.