Among the relations S = { ( a , b ) : a , b R - { 0 } , 2 + a b 0 } and T = { ( a , b ) : a , b R , a 2 - b 2 Z } [2023]

Question

Among the relations

$S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and

$T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ [2023]

Smart Achievers · Accepted Answer

(4)

For relation $T = a^{2} - b^{2} \in I$

Then, $(b, a)$ on relation $R \Rightarrow b^{2} - a^{2} \in I$

$∴ T is symmetric,$ $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$

$2 + \frac{a}{b} > 0 \Rightarrow \frac{a}{b} > - 2 \Rightarrow \frac{b}{a} > - \frac{1}{2}$

If $(b, a) \in S$ , then $2 + \frac{b}{a}$ is not necessarily positive.

$∴$ $S is not symmetric.$

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