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Relations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations4 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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