Q. 104.7( 6 Votes )

Let A be the set of all points in a plane and let O be the origin. Show that the relation R = {(P, Q) : P, Q A and OP = OQ) is an equivalence relation.

Answer :

In order to show R is an equivalence relation, we need to show R is Reflexive, Symmetric and Transitive.


Given that, A be the set of all points in a plane and O be the origin. Then, R = {(P, Q) : P, Q A and OP = OQ)}


Now,


R is Reflexive if (P,P) R P A


P A , we have


OP=OP


(P,P) R


Thus, R is reflexive.


R is Symmetric if (P,Q) R (Q,P) R P, Q A


Let P, Q A such that,


(P,Q) R


OP = OQ


OQ = OP


(Q,P) R


Thus, R is symmetric.


R is Transitive if (P,Q) R and (Q,S) R (P,S) R P, Q, S A


Let (P,Q) R and (Q,S) R P, Q, S A


OP = OQ and OQ = OS


OP = OS


(P,S) R


Thus, R is transitive.


Since R is reflexive, symmetric and transitive it is an equivalence relation on A.


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