Let $R$ be the set of real numbers. Statement I : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$. Statement II : $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equivalence relation on $R$.

Question

Let $R$ be the set of real numbers.

Statement I : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$.

Statement II : $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equivalence relation on $R$.

Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.

Statement I is true, Statement II is false.

Statement I is false, Statement II is true.

Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.

Solution

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