#### Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 11 Maths Textbook Solution.

Answer: R is an equivalence relation on A
Hint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.
$Given: O\; be\; the\; origin\; and\; R=\left \{ \left (P, Q \right ):OP=OQ \right \} be\; a\; relation \; on\; A \; where\; O\; is \; the\; origin$.
Explanation:
Let A be a set of points on a plane.
Reflexivity:
$Let\: P \: \epsilon \: A$
$Since, OP=OP=(P, P) \: \epsilon \: R$
R is reflexive.
Symmetric:
$Let (P, Q) \: \epsilon \: R \: f\! or\: P, Q \: \epsilon \: A$
$Then \: OP=OQ$
$OQ=OP$
$(Q, P) \: \epsilon \: R$
R is symmetric
Transitive:
$Let (P, Q) \: \epsilon \: R\: and\: (Q, S) \: \epsilon \: R$
$OP=OQ \: \: ...(i)\: and \: \: OQ=OS \: \:....(ii)$
Putting  (ii) in (i), we get
$OP=OS$
$(P, S)\: \epsilon \: R$
R is transitive
Therefore, R is reflexive, symmetric and transitive.
Thus, R is an equivalence relation on A.