Let R be the set of real numbers.

end{enumerate}

Statement-1: A = {(x,y) in R X R:y - x is an integer }

is an equivalence . relation on R.

Statement-2: B = {(x,y) in R X R:x = alpha y for some

rational number } is . an equivalence relation on R.

Question

Let R be the set of real numbers. Statement-1: A = \(x,y) in R x R:y - x is an integer \ is an equivalence . relation on R.\ Statement-2: B = \(x,y) in R x R:x = alpha y for some rational number \ is . an equivalence relation on R.

DivineJEE · Accepted Answer

for Statement-1.\ (d) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct . explanation for Statement-1.\ Solution: 38. Explanation : - x-y is an integer - x-x=0 is an integer => A is reflexive - x-y is an integer => y-x is an integer => A is symmetric - x-y, y-z are integers As sum of two integers is an integer. Therefore, (x-y)+(y-z)=(x-z) is an integer, which implies that A is transitive. Hence, Statement-1 is true. Also, - x / x=1 is a rational number ⇒ B is reflexive - x / y=alpha is rational ⇒ y / x need not be rational, that is, 0 / 1 is rational => 1 / 0 is not rational Hence, B is not symmetric, that is, B is not an equivalence relation.

Explanation

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