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12 Show that the oulation \( R \) on the set \( A \) of points on co-ordinate plane givem by \( R=\{(P, Q) \) : idistance \( O P=O Q \), where \( O \) is the origin ) is an equivalence velation. Furtier find the set of als points velated Uto apoint \( P \neq(0,0) \) is the circlepeissing throrgh \( p \) with the rigin as centre. |
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Answer» A is the set of points in a plane. R = {(P. Q): distance of the point P from the origin is same as the distance of the point Q from the origin} = {(PQ) |OP| = |OQ | where O is origin} Since | OP | = | OPI, (PP) ERVPE A. ... R is reflexive. Also (P. Q) ER ⇒|OP| = |OQ| ⇒ | OQ | = |OP| ⇒ (Q.P) ER⇒ R is symmetric. Next let (PQ) E R and (Q, T) ER⇒ |OP|=|OQ | and OQ | = |OT| ⇒ |OP| = |OT| →> (PT) ER .. R is transitive. ... R is an equivalence relation. Set of points related to P = 0 = {Q E A: (Q,P) E R} = {QEA: |OQ| = |OP|} = {Q E A :Q lies on a circle through P with centre O}.
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