Q. 104.0( 7 Votes )

Show that the relation R, defined on the set A of all polygons as

R = {(P1, P2) : P1 and P2 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?

Answer :

R = {(P1, P2): P1 and P2 have same the number of sides}

Proof :


To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.


Reflexivity: For Reflexivity, we need to prove that-


(a, a) R


R is reflexive since (P1, P1) R as the same polygon has the same number of sides with itself.


Symmetric: For Symmetric, we need to prove that-


If (a, b) R, then (b, a) R


Let (P1, P2) R.


P1 and P2 have the same number of sides.


P2 and P1 have the same number of sides.


(P2, P1) R


R is symmetric.


Transitive: For Transitivity, we need to prove that-


If (a, b) R and (b, c) R, then (a, c) R


Now, (P1, P2), (P2, P3) R


P1 and P2 have the same number of sides. Also, P2 and P3 have the same number of sides.


P1 and P3 have the same number of sides.


(P1, P3) R


R is transitive.


Hence, R is an equivalence relation.


And, now the elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have three sides (since T is a polygon with three sides).


Hence, the set of all elements in A related to triangle T is the set of all triangles.


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