Let X=R × R . Define a relation R on X as : (a_1, b_1) R(a_2, b_2) b_1=b_2 Statement I : R is an equivalence relation. Statement II : For some (a, b) ∈ X , the set S=\ (x, y) ∈ X:(x, y) R(a, b)\ represents a line

Correct answer: B. Statement I is true but Statement II is false Reflexive : (a_1, b) R(a_1, b_1) b_1=b_1 True array rl Symmetric : & (a_1, b_1) R (a_2, b_2) b_1=b_2 & (a_2, b_2) R (a_1, b_1) b_2=b_1 array True aligned Transitive : & (a_1, b_1) R(a_2, b_2) b_1=b_2 & &(a_2, b_2) R(a_3, b_3) b_2=b_3 & (a_1, b_1) R(a_3 . b_3) True b_1=b_3 aligned Hence Relation R is an equivence relation Statement-I is true. For statement - II y=b so False

JEE Main 202523 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 23 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Sets and Relations

JEE Main 2025 · 23 January · Evening Shift · Mathematics · Sets and Relations

Let X=R × R . Define a relation R on X as : (a_1, b_1) R(a_2, b_2)...

Let

X=\mathbf{R} \times \mathbf{R}

. Define a relation

R

X

as :

\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2

Statement I :

\quad \mathrm{R}

is an equivalence relation. Statement II : For some

(a, b) \in X

, the set

S=\{(x, y) \in X:(x, y) R(a, b)\}

represents a line parallel to

y=x

. In the light of the above statements, choose the correct answer from the options given below :

Options

ABoth Statement I and Statement II are false
BStatement I is true but Statement II is false
CBoth Statement I and Statement II are true
DStatement I is false but Statement II is true

Correct answer

B. Statement I is true but Statement II is false

Step-by-step solution

Reflexive :

\left(a_1, b\right) R\left(a_1, b_1\right) \Rightarrow b_1=b_1 \quad

True

\begin{array}{rl}\text{ Symmetric : } & \left(a_1, b_1\right) R \left(a_2, b_2\right) \Rightarrow b_1=b_2 \\ & \left(a_2, b_2\right) R \left(a_1, b_1\right) \Rightarrow b_2=b_1\end{array}

True

\begin{aligned} \text{ Transitive : } & \left(a_1, b_1\right) R\left(a_2, b_2\right) \Rightarrow b_1=b_2 \\ & &\left(a_2, b_2\right) R\left(a_3, b_3\right) b_2=b_3 \\ & \Rightarrow\left(a_1, b_1\right) R\left(a_3 . b_3\right) \Rightarrow \text{ True } b_1=b_3\end{aligned}

Hence Relation

R

is an equivence relation Statement-I is true. For statement -

\mathrm{II}

\Rightarrow \mathrm{y}=\mathrm{b} so False

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