Question

Board & Competitive Exams (Level)
6.
Relations and Functions
4.
Let O be the origin. If a relation R is defined between two points A and B in a plane such that OA = og
Then the relation Ris
(1) Reflexive only
(2) Symmetric only
(3) Transitive only
(4) Equivalence
​

Answer 1

Appropriate Question :-

Let O be the origin. If a relation R is defined between two points A and B in a plane such that OA = OB.

Then the relation R is

• (1) Reflexive only

• (2) Symmetric only

• (3) Transitive only

• (4) Equivalence

Solution :-

Let O be the origin in the given plane.

Then relation R is defined as

• R = {(P, Q) : OP = OQ}

Case :- 1 Reflexive

For any point P in the plane, we know

☆ OP = OP

⇛ (P, P) ∈ R

⇛ R is Reflexive.

Case :- 2 Symmetric

Let assume that P and Q be points in a plane such that

☆ (P, Q) ∈ R

⇛ OP = OQ

⇛ OQ = OP

⇛ (Q, P) ∈ R

⇛ R is Symmetric.

Case :- 3 Transitive

Let assume that P, Q, R be points in a plane such that

☆ (P, Q) ∈ R and (Q, R) ∈ R

⇛ OP = OQ and OQ = OR

⇛ OP = OR

⇛ (P, R) ∈ R

⇛ R is Transitive.

Since, R is Reflexive, Symmetric and Transitive

⇛ R is an equivalence relation.

Hence,

$\: \: \: \: \: \: \: \: \: \: \large{\boxed{\boxed{\bf{Option \: (4) \: is \: correct}}}}$