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Let R be the set of real numbers. Statement-1: A={(x,y)∈R×R : x−y is an integer} is an equivalence relation on R. Statement-2: B={(x,y)∈R×R : x=y for some rational number α} is an equivalence relation on R.

1. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
2. Statement-1 is true, Statement-2 is false.
3. Statement-1 is false, Statement-2 is true.
4. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Correct answer: Statement-1 is false, Statement-2 is true.

Solution

Statement-1 is false because the relation defined does not satisfy the properties of reflexivity, symmetry, and transitivity for all real numbers. Statement-2 is true as it correctly defines an equivalence relation where two numbers are considered equivalent if they are equal, fulfilling all necessary properties.

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