Mark the correct alternative in the following question: For real numbers x and y, define xRy if x-y+sqrt2 is an irrational number. Then the relation R is ___________ .

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KollegeApply · Accepted Answer

We have, R = {(x, y) : x−y+sqrt2 is an irrational number; x, y ∈ R} As, x−x+sqrt2 = sqrt2, which is an irrational number ⇒ (x, x) ∈ R So, R is reflexive relation Since, (sqrt2, 2) ∈ R i.e. sqrt2−2+sqrt2=2sqrt2−2, which is an irrational number but 2−sqrt2+sqrt2=2, which is a rational number ⇒ (2, sqrt2) ∉ R So, R is not symmetric relation Also, (sqrt2, 2) ∈ R and (2, 2sqrt2) ∈ R i.e. sqrt2−2+sqrt2=2sqrt2−2, which is an irrational number and 2−2sqrt2+sqrt2=2−sqrt2, which is also an irrational number But sqrt2−2sqrt2+sqrt2=0, which is a rational number ⇒ (sqrt2, 2sqrt2) ∉ R So, R is not transitive relation Hence, R is Reflexive.

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