Let $X=\mathbf{R} \times \mathbf{R}$. Define a relation $R$ on $X$ as : $\left(a_1, b_1\right) R\left(a_2,…

Let $X=\mathbf{R} \times \mathbf{R}$. Define a relation $R$ on $X$ 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 :
  1. Both Statement I and Statement II are false
  2. Statement I is true but Statement II is false
  3. Both Statement I and Statement II are true
  4. Statement I is false but Statement II is true

Solution

Reflexive : $\left(a_1, b\right) R\left(a_1, b_1\right) \Rightarrow b_1=b_1 \quad$ True
$\left.\begin{array}{rl}\text { Symmetric : } & \left(a_1, b_1\right) R\left(a_2, b_2\right) \Rightarrow b_1=b_2 \\ & \left(a_2, b_2\right) R\left(a_1, b_1\right) \Rightarrow b_2=b_1\end{array}\right\}$ 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 ^

Asked in: JEE Main 2025 (23 Jan Shift 2)

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