Let R be the set of real numbers
Statement-1: $A = {(x, y) \in R \times R : y + x$ is an integer} is an equivalence relation on R.
Statment-2: $B = {(x, y) \in R \times R : y = α x$ for some rational number $α$ } is not equivalence relation on R.

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STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

answer is D.

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Detailed Solution

A is not reflexive as x + x may not be integer " $x \in R$ . So statement-1 is false

Statememt-2 is true as B is not symmetric, because $(\sqrt{3}, 0)$ $\in B as 0 = \sqrt{3} \times 0 for α = 0, but (0, \sqrt{3}) \notin B .$

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