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Let O be the origin. We define a relation between two points P

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Let $O$ be the origin. We define a relation between two points $P$ and $Q$ in a plane, if $OP = OQ$ . Show that the relation, so defined is an equivalence relation.

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Levi Discussed

Let

O

be the origin. We define a relation between two points

P

and

Q

in a plane, if

OP = OQ

. Show that the relation, so defined is an equivalence relation.

15 mins ago

Discuss this question LIVE

15 mins ago

Text solutionVerified

Let us denote this relation by

R = {(P, Q) : OP = OQ}

for

O

being the origin.

Now

(P, P) \in R

since

OP = OQ

for any point

P

. So the relation is reflexive.

Again this relation is symmeric as if

(P, Q) \in R \Rightarrow (Q, P) \in R

since

OP = OP \Rightarrow OQ = OP

for all

P, Q

Also this relation is transitive as if

(P, Q) \in R, (Q, S) \in R \Rightarrow (P, S) \in R

since

OP = OQ, OQ = OS \Rightarrow OP = OS

for all

P, Q, S

Hence the relation is reflexive, symmetric and transitive so it is an equivalance relation.

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Question Text	Let $O$ be the origin. We define a relation between two points $P$ and $Q$ in a plane, if $OP = OQ$ . Show that the relation, so defined is an equivalence relation.
Answer Type	Text solution:1
Upvotes	150

Let O be the origin. We define a relation between two points P and Q in a plane, if OP=OQ. Show that the relation, so defined is an equivalence relation.

Text solutionVerified

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Let $O$ be the origin. We define a relation between two points $P$ and $Q$ in a plane, if $OP = OQ$ . Show that the relation, so defined is an equivalence relation.