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: \quad R is an equivalence relation. Statement II : For some (a, b) \in X , the \operatornameset 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 :

Statement I is true but Statement II is false

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is false but Statement II is true

Sets and Relations PYQ — JEE Main 2025 · 23 Jan · Shift 2 · Q32 with Solution

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Correct answer: B

Given relation

On $X=\mathbb{R}\times \mathbb{R}$ , define

(a_1,b_1)\,R\,(a_2,b_2) \iff b_1=b_2.

So two ordered pairs are related iff their second coordinates are equal.

Check Statement I: $R$ is an equivalence relation

To be an equivalence relation, $R$ must be reflexive, symmetric, and transitive.

(i) Reflexive

For any $(a,b)\in X$ , we have

b=b.

Hence

(a,b)R(a,b).

So $R$ is reflexive.

(ii) Symmetric

Suppose

(a_1,b_1)R(a_2,b_2).

Then by definition,

b_1=b_2.

Therefore,

b_2=b_1,

which implies

(a_2,b_2)R(a_1,b_1).

So $R$ is symmetric.

(iii) Transitive

Suppose

(a_1,b_1)R(a_2,b_2) \quad \text{and} \quad (a_2,b_2)R(a_3,b_3).

Then

b_1=b_2 \quad \text{and} \quad b_2=b_3.

Hence

b_1=b_3,

(a_1,b_1)R(a_3,b_3).

Thus $R$ is transitive.

Therefore, $R$ is an equivalence relation.

So Statement I is true.

Check Statement II

For some $(a,b)\in X$ , consider

S=\{(x,y)\in X:(x,y)R(a,b)\}.

By definition of $R$ ,

(x,y)R(a,b) \iff y=b.

Therefore,

S=\{(x,y)\in \mathbb{R}^2 : y=b\}.

This is a horizontal line parallel to the $x$ -axis.

A line parallel to $y=x$ must have slope $1$ , i.e. be of the form

y=x+c.

But $y=b$ has slope $0$ , so it is not parallel to $y=x$ .

Hence Statement II is false.

Conclusion

Statement I: True
Statement II: False

Therefore the correct option is:

\boxed{\text{B}}

More from Sets and Relations

View written solutionFree

Correct answer: B

Given relation

On $X=\mathbb{R}\times \mathbb{R}$ , define

(a_1,b_1)\,R\,(a_2,b_2) \iff b_1=b_2.

So two ordered pairs are related iff their second coordinates are equal.

Check Statement I: $R$ is an equivalence relation

To be an equivalence relation, $R$ must be reflexive, symmetric, and transitive.

(i) Reflexive

For any $(a,b)\in X$ , we have

b=b.

Hence

(a,b)R(a,b).

So $R$ is reflexive.

(ii) Symmetric

Suppose

(a_1,b_1)R(a_2,b_2).

Then by definition,

b_1=b_2.

Therefore,

b_2=b_1,

which implies

(a_2,b_2)R(a_1,b_1).

So $R$ is symmetric.

(iii) Transitive

Suppose

(a_1,b_1)R(a_2,b_2) \quad \text{and} \quad (a_2,b_2)R(a_3,b_3).

Then

b_1=b_2 \quad \text{and} \quad b_2=b_3.

Hence

b_1=b_3,

(a_1,b_1)R(a_3,b_3).

Thus $R$ is transitive.

Therefore, $R$ is an equivalence relation.

So Statement I is true.

Check Statement II

For some $(a,b)\in X$ , consider

S=\{(x,y)\in X:(x,y)R(a,b)\}.

By definition of $R$ ,

(x,y)R(a,b) \iff y=b.

Therefore,

S=\{(x,y)\in \mathbb{R}^2 : y=b\}.

This is a horizontal line parallel to the $x$ -axis.

A line parallel to $y=x$ must have slope $1$ , i.e. be of the form

y=x+c.

But $y=b$ has slope $0$ , so it is not parallel to $y=x$ .

Hence Statement II is false.

Conclusion

Statement I: True
Statement II: False

Therefore the correct option is:

\boxed{\text{B}}

Sets and Relations question

Sets and Relations question

(i) Reflexive

(ii) Symmetric

(iii) Transitive

More from Sets and Relations

Sets and Relations question

Sets and Relations question

(i) Reflexive

(ii) Symmetric

(iii) Transitive

More from Sets and Relations