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Relations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRelations5 Marks
Question
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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Answer

Let A be set of points on plane. Let R = {(P, Q): OP = OQ} be a relation on A where O is the origin. To prove R is an equivalence relation, we need to show that R is reflexive, symmetric and transitive on A. Now, Reflexivity: Let $\text{P}\in\text{A}$ Since OP = OP $\Rightarrow\ (\text{P, P})\in\text{R}$ ⇒ R is reflexive. Symmetric: Let $(\text{P, Q})\in\text{R}$ for $\text{P, Q}\in\text{A}$ Then OP = OQ ⇒ OQ = OP$\Rightarrow\ (\text{Q, P})\in\text{R}$
⇒ R is symmetric. Transitive: Let $(\text{P, Q})\in\text{R}$ and $(\text{Q, S})\in\text{R}$ ⇒ OP = OQ and OQ = OS ⇒ OP = OS $\Rightarrow\ (\text{P, S})\in\text{R}$ ⇒ R is transitive. Thus, R is an equivalence relation on A.

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