Show 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}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

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Solution

The given relation R in the set A of all polygons is defined as R={ ( P 1 , P 2 ): P 1 and P 2 have same number of sides }.

Every polygon has the same number of sides as itself.

⇒( P 1 , P 1 )∈R

So R is reflexive.

Let, ( P 1 , P 2 )∈R,

⇒ P 1 and P 2 have same no of sides ⇒ P 2 and P 1 have same no of sides ⇒( P 2 , P 1 )∈R

So, Ris symmetric.

Let, ( P 1 , P 2 )and ( P 2 , P 3 )∈R.

⇒ P 1 and P 2 have same number of sides.

Also, P 2 and P 3 have same number of sides.

⇒ P 1 and P 3 have same number of sides.

So, ( P 1 , P 3 )∈R. Hence, Ris transitive.

Thus, R is an equivalence relation.

The given right triangle is T 1 with sides ( 3,4,5 ).

The set of all elements in A related to the given right triangle consist of those polygons which have 3 sides. Thus, the elements of set A related to triangle T is set of all triangles.

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