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 T, then R is:

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 $a, b \in T$ , then $R$ is:

Smart Achievers · Accepted Answer

(3)

Ans. Equivalence

Explanation:

Each triangle is congruent to itself.

$∴$ Relation is reflexive.

Suppose $∆ A B C ≅ ∆ P Q R$

then $∆ P Q R ≅ ∆ A B C$

$∴$ Relation is symmetric.

Suppose $∆ A B C ≅ ∆ P Q R,$

and $∆ P Q R ≅ ∆ X Y Z$

then $∆ A B C ≅ ∆ X Y Z$

$∴$ Relation is transitive.

$∴$ Relation is an equivalence relation.

Solution

Q.
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 $a, b \in T$ , then $R$ is:

1 reflexive but not symmetric

2 transitive but not symmetric

3 equivalence

4 neither symmetric nor transitive

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