Class JEE Mathematics Sets, Relations, and Functions Q #1010
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4 Marks 2025 JEE Main 2025 (Online) 23rd January Evening Shift MCQ SINGLE
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:
(A) Both Statement I and Statement II are true
(B) Statement I is true but Statement II is false
(C) Both Statement I and Statement II are false
(D) Statement I is false but Statement II is true
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Correct Answer: B
Explanation
Statement I: Reflexive: $(a_1, b_1) R (a_1, b_1) \Rightarrow b_1 = b_1$ (True). Symmetric: $(a_1, b_1) R (a_2, b_2) \Rightarrow b_1 = b_2$ and $(a_2, b_2) R (a_1, b_1) \Rightarrow b_2 = b_1$ (True). Transitive: $(a_1, b_1) R (a_2, b_2) \Rightarrow b_1 = b_2$ and $(a_2, b_2) R (a_3, b_3) \Rightarrow b_2 = b_3$. Thus, $b_1 = b_3$, so $(a_1, b_1) R (a_3, b_3)$ (True). Hence, relation $R$ is an equivalence relation, and Statement I is true. For Statement II: $(x, y) R (a, b) \Rightarrow y = b$. This represents a line $y=b$, which is parallel to the x-axis. It is not parallel to the line $y=x$. Therefore, Statement II is false.

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