Q. 133.9( 67 Votes )

# Show that the rel

Answer :

It is given that the relation R defined in the set A of all polygons as

R = {(P_{1}, P_{2}): P_{1} and P_{2} have same number of sides},

Then, R is reflexive since (P_{1}, P_{2}) ϵ R as the same polygon has the same number of sides with itself.

Let (P_{1}, P_{2}) ϵ R

⇒ P_{1} and P_{2} have the same number of sides.

⇒ P_{2} and P_{1} have the same number of sides.

⇒ (P_{2}, P_{1}) ϵ R

Therefore, R is symmetric.

Now, let (P_{1}, P_{2}), (P_{2}, P_{3}) ϵ R

⇒ P_{1} and P_{2} have the same number of sides. Also, P_{2} and P_{3} have the same number of sides.

⇒ P_{1} and P_{3} have the same number of sides.

⇒ (P_{1}, P_{3}) ϵ R

Therefore, R is transitive.

Thus, R is an equivalence relation.

The elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have 3 sides.

Therefore, the set of all elements in A related to triangle T is the set of all triangels.

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