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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