JEE Main 2025 Jan 23 Shift 2, Mathematics Q14: Relations

JEE Main2025Jan 23, Shift 2Mathematics

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 false.
B
Statement-I is true but Statement-II is false.
C
Both Statement-I and Statement-II are true.
D
Statement-I is false but Statement-II is true.

Solution

Statement-I:

Reflexive: $b_{1} = b_{1}$ always, so $(a_{1}, b_{1}) R (a_{1}, b_{1})$ . $✓$

Symmetric: $b_{1} = b_{2} \Rightarrow b_{2} = b_{1}$ . $✓$

Transitive: $b_{1} = b_{2}$ and $b_{2} = b_{3} \Rightarrow b_{1} = b_{3}$ . $✓$

So $R$ is an equivalence relation. Statement-I is true.

Statement-II: The equivalence class of $(a, b)$ is $S = {(x, y) : y = b}$ , which is the horizontal line $y = b$ . Horizontal lines have slope $0$ and are parallel to $y = 0$ (the $x$ -axis), not to $y = x$ (slope $1$ ). Statement-II is false.

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JEE Main 2025 Jan 23 Shift 2, Mathematics Q14: Relations

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