JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Sets Relations and Functions

Let

X = R \times R

. Define a relation

R

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:

Answer: (B)

Solution

Check the three equivalence properties for

R

X = R \times R

defined by

(a_{1}, b_{1}) R (a_{2}, b_{2}) \Leftrightarrow b_{1} = b_{2}

:

Reflexive:

b_{1} = b_{1}

always holds.
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}

.

All three hold, so

R

is an equivalence relation — Statement-I is true.

For a fixed

(a, b)

, the set

S = {(x, y) : (x, y) R (a, b)} = {(x, y) : y = b}

is the horizontal line

y = b

, which is parallel to the x-axis, not to

y = x

. Statement-II is false.

Statement-I is true but Statement-II is false.

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Single correct option