Let $\boldsymbol{X}=\mathbf{R} \times \mathbf{R}$. 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: $\mathbf{R}$ 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:

Step 1: Analyze Statement I for Equivalence Relation Properties

To determine if $R$ is an equivalence relation, we must check for reflexivity, symmetry, and transitivity. A relation $R$ on a set $X$ is reflexive if $(a,a) \in R$ for every $a \in X$. It is symmetric if whenever $(a,b) \in R$, then $(b,a) \in R$. It is transitive if whenever $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$. In our case, the relation is defined as $(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1=b_2$. We can see that $b=b$ is always true, so it's reflexive. If $b_1=b_2$, then $b_2=b_1$, so it's symmetric. If $b_1=b_2$ and $b_2=b_3$, then $b_1=b_3$, so it's transitive. Since all three properties hold, $R$ is an equivalence relation.

Step 2: Analyze Statement II for the set S

Statement II describes a set $S$ based on the relation $R$. Given $(a, b) \in X$, the set $S$ consists of all points $(x, y)$ such that $(x, y) R (a, b)$. According to the definition of $R$, this means $y = b$. Therefore, the set $S$ represents all points $(x, y)$ where the $y$ -coordinate is fixed at $b$, while $x$ can be any real number. This describes a horizontal line.

Step 3: Compare the line S with $y=x$

The set $S$ represents a horizontal line with equation $y=b$. A line parallel to $y=x$ would have a slope of $1$. A horizontal line has a slope of $0$. Since the slopes are different ($0 \neq 1$), a horizontal line $y=b$ is not parallel to the line $y=x$. Therefore, Statement II is false.

Step 4: Conclusion

Based on our analysis, Statement I, which claims $R$ is an equivalence relation, is true. Statement II, which claims the set $S$ represents a line parallel to $y=x$, is false because $S$ represents a horizontal line $y=b$, which is not parallel to $y=x$.

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