Let X = R R. Define a relation R on X as: . Statement I: R is an equivalence relation. Statement II: For some (a, b) X, the set S = {(x, y) X :(x, y) R (a, b)} represents a line parallel to y = x. In the light of the above statements, choose the c

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Solution

Q.
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: [2025]

1 Statement I is true but Statement II is false.

2 Statement I is false but State II is true.

3 Both Statement I and Statement II are false.

4 Both Statement I and Statement II is true.

Ans.
(1)

Statement I:

Reflexive : $(a_{1}, b_{1}) R (a_{1}, b_{1}) \Rightarrow b_{1} = b_{1}$

$∴$ R is reflexive.

Symmetric : $(a_{1}, b_{1}) R (a_{2}, b_{2}) \Rightarrow b_{1} = b_{2}$

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

$∴$ R is symmetric.

Transitive : $(a_{1}, b_{1}) R (a_{2}, b_{2}) \Rightarrow b_{1} = b_{2}$

and $(a_{2}, b_{2}) R (a_{3}, b_{3}) \Rightarrow b_{2} = b_{3} \Rightarrow b_{1} = b_{3}$

$\Rightarrow (a_{1}, b_{1}) R (a_{3}, b_{3})$

Hence, relation R is equivalence relation.

$∴$ Statement I is true.

For Statement II: (x, y) R (a, b), for some $(a, b) \in X \Rightarrow y = b$

So, Statement II is false.

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