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

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Q.
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]

1 S is transitive but T is not

2 both S and T are symmetric

3 neither S nor T is transitive

4 T is symmetric but S is not

Ans.
(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.

$∴$ $S is not symmetric.$

Solution

Q.
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]

1 S is transitive but T is not

2 both S and T are symmetric

3 neither S nor T is transitive

4 T is symmetric but S is not

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Solution

Q. Among the relations S={(a,b):a,b∈R-{0},2+ab>0} and T={(a,b):a,b∈R,a2-b2∈Z} [2023]

1 S is transitive but T is not

2 both S and T are symmetric

3 neither S nor T is transitive

4 T is symmetric but S is not

Ans. (4) For relation T=a2-b2∈I Then, (b,a) on relation R⇒b2-a2∈I ∴T is symmetric, S={(a,b):a,b∈R-{0}, 2+ab>0} 2+ab>0⇒ab>-2⇒ba>-12 If (b,a)∈S, then 2+ba is not necessarily positive. ∴ S is not symmetric.

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Q.
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]