Let $$X=R\times R$$ Define a relation R on X as : $$(a_{1},b_{1})R(a_{2},b_{2}) \Leftrightarrow b_{1}=b_{2}$$ Statement I : 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 :

We need to evaluate two statements about the relation R on $$X = \mathbb{R} \times \mathbb{R}$$.

Relation: $$(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1 = b_2$$

Statement I: R is an equivalence relation.

Check reflexive: $$(a,b) R (a,b)$$ since $$b = b$$. TRUE.

Check symmetric: If $$(a_1,b_1) R (a_2,b_2)$$, then $$b_1 = b_2$$, so $$b_2 = b_1$$, hence $$(a_2,b_2) R (a_1,b_1)$$. TRUE.

Check transitive: If $$(a_1,b_1) R (a_2,b_2)$$ and $$(a_2,b_2) R (a_3,b_3)$$, then $$b_1 = b_2$$ and $$b_2 = b_3$$, so $$b_1 = b_3$$, hence $$(a_1,b_1) R (a_3,b_3)$$. TRUE.

Statement I is TRUE.

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.

$$S = \{(x,y) : y = b\}$$

This is a horizontal line (parallel to the x-axis), not a line parallel to y = x.

The line y = x has slope 1. The line y = b has slope 0. These are not parallel for any value of b.

Statement II is FALSE.

The correct answer is Option 2: Statement I is true but Statement II is false.