Let a set \(\mathrm{A}=\mathrm{A}_{1} \cup \mathrm{~A}_{2} \cup \ldots \cup \mathrm{~A}_{k}\), where \(A_{i} \cap A_{j}=\phi\) for \(i \neq j, 1 \leq i, j \leq k\). Define the relation \(R\) from \(A\) to \(A\) by \(R=\left\{(x, y): y \in A_{i}\text{ if and only if }x \in A_{i}, 1 \leq i \leq k\right\}\). Then, R is :

(A) reflexive, symmetric but not transitive

(B) reflexive, transitive but not symmetric

(C) reflexive but not symmetric and transitive

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Sets Relations and Functions

Let a set

A = A_{1} \cup A_{2} \cup \dots \cup A_{k}

, where

A_{i} \cap A_{j} = ϕ

for

i \neq = j, 1 \leq i, j \leq k

. Define the relation

R

from

A

A

R = {(x, y) : y \in A_{i} if and only if x \in A_{i}, 1 \leq i \leq k}

. Then, R is :

Answer: (D)

The relation

R

relates elements that belong to the same subset

A_{i}

.

Reflexive: For any

x \in A

x

belongs to some

A_{i}

, so

(x, x) \in R

. ✓

Symmetric: If

(x, y) \in R

, then both

x, y \in A_{i}

for some

i

. This means

(y, x) \in R

. ✓

Transitive: If

(x, y) \in R

and

(y, z) \in R

, then

x, y \in A_{i}

and

y, z \in A_{j}

. Since the sets are disjoint and

y

is in both, we must have

i = j

, so

x, z \in A_{i}

, meaning

(x, z) \in R

. ✓

Therefore,

R

is an equivalence relation.

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Single correct option