Let $\mathrm{X}=\mathbf{R} \times \mathbf{R}$. 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 $(\mathrm{a}, \mathrm{b}) \in \mathrm{X}$, the $\operatorname{set} \mathrm{S}=\{(x, y) \in \mathrm{X}:(x, y) \mathrm{R}(\mathrm{a}, \mathrm{b})\}$ represents a line parallel to $y=x$.

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

<p>Statement - I :</p> <p>Reflexive : $\left(a_1, b_1\right) R\left(a_1, b_1\right) \Rightarrow b_1=b_1 \quad$ True</p> <p>$\left.\begin{array}{rl}\text { Symmetric : } & \left(\mathrm{a}_1, \mathrm{~b}_1\right) \mathrm{R}\left(\mathrm{a}_2, \mathrm{~b}_2\right) \Rightarrow \mathrm{b}_1=\mathrm{b}_2 \\ & \left(\mathrm{a}_2, \mathrm{~b}_2\right) \mathrm{R}\left(\mathrm{a}_1, \mathrm{~b}_1\right) \Rightarrow \mathrm{b}_2=\mathrm{b}_1\end{array}\right\}$ True</p> <p>$\left.\begin{array}{ll}\text { Transitive: } & \left(a_1, b_1\right) R\left(a_2, b_2\right) \Rightarrow b_1=b_2 \\ & \&\left(a_2, b_2\right) R\left(a_3, b_3\right) b_2=b_3\end{array}\right\} b_1=b_3$</p> <p>$$\Rightarrow\left(\mathrm{a}_1, \mathrm{~b}_1\right) \mathrm{R}\left(\mathrm{a}_3 \cdot \mathrm{~b}_3\right) \Rightarrow \text { True }$$</p> <p>Hence Relation $R$ is an equivence relation Statement-I is true.</p> <p>For statement - II $\Rightarrow \mathrm{y}=\mathrm{b}$ so False</p>