Q. 28

# 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

A. reflexive but not transitive

B. transitive but not symmetric

C. equivalence

D. none of these

Answer :

Given that,

R be a relation on T defined as aRb if a is congruent to b ∀ a, b ∈ T

Now,

aRa ⇒ a is congruent to a, which is true since every triangle is congruent to itself.

⇒ (a,a) ∈ R ∀ a ∈ T

⇒ R is reflexive.

Let aRb ⇒ a is congruent to b

⇒ b is congruent to a

⇒ bRa

∴ (a,b) ∈ R ⇒ (b,a) ∈ R ∀ a, b ∈ T

⇒ R is symmetric.

Let aRb ⇒ a is congruent to b and bRc ⇒ b is congruent to c

⇒ a is congruent to c

⇒ aRc

∴ (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R ∀ a, b,c ∈ T

⇒ R is transitive.

Hence, R is an equivalence relation.

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Fill in the blanks in each of the

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.

Mathematics - ExemplarFill in the blanks in each of the

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a^{2} – b^{2}| < 8}. Then R is given by _______.

State True or False for the statements

Every relation which is symmetric and transitive is also reflexive.

Mathematics - ExemplarState True or False for the statements

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

Mathematics - Exemplar