Let R be a relation on real numbers given by is an irrational number. Then R is

Question

Let R be a relation on real numbers given by $R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $}$ . Then R is

Smart Achievers · Accepted Answer

(1)

$Check for reflexivity: As 3 (a - a) + \sqrt{7} = \sqrt{7} which belongs to relation$

$So relation is reflexive.$

$Check for symmetric$

$Take a = \frac{\sqrt{7}}{3}, b = 0 Now (a, b) \in R but (b, a) \notin R$

$As 3 (b - a) + \sqrt{7} = 0 which is rational so relation is not symmetric$

$Check for Transitivity, Take (a, b) as (\frac{\sqrt{7}}{3}, 1) & (b, c) as (1, \frac{2 \sqrt{7}}{3})$

$So now (a, b) \in R & (b, c) \in R but (a, c) \notin R,$

$Which means relation is not transitive$

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