Define a relation R on the interval by xRy if and only if . Then R is : [2025]

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Q.
Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

1 an equivalence relation.

2 reflexive but neither symmetric not transitive.

3 both reflexive and symmetric but not transitive.

4 both reflexive and transitive but not symmetric.

Ans.
(1)

$\sec^{2} x - \tan^{2} x = 1, \forall x \in [0, \frac{π}{2})$

$∴$ R is reflexive

Consider, $\sec^{2} x - \tan^{2} y = 1$

$\Rightarrow 1 + \tan^{2} x - (\sec^{2} y - 1) = 1$

$\Rightarrow 1 + \tan^{2} x - s e c^{2} y + 1 = 1 \Rightarrow \sec^{2} y - \tan^{2} x = 1$

$∴$ R is symmetric.

Now, if $\sec^{2} x - \tan^{2} y = 1$ and $\sec^{2} y - \tan^{2} z = 1$

Adding both equation, $\sec^{2} x - \tan^{2} y + \sec^{2} y - \tan^{2} z = 2$

$\Rightarrow \sec^{2} x - \tan^{2} z = 1$ [ $∵ \sec^{2} y - \tan^{2} y = 1$ ]

$∴$ R is transitive

Thus R is an equivalence relation.

Solution

Q.
Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

1 an equivalence relation.

2 reflexive but neither symmetric not transitive.

3 both reflexive and symmetric but not transitive.

4 both reflexive and transitive but not symmetric.

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Solution

Q. Define a relation R on the interval [0,π2) by xRy if and only if sec2x–tan2y=1. Then R is : [2025]

1 an equivalence relation.

2 reflexive but neither symmetric not transitive.

3 both reflexive and symmetric but not transitive.

4 both reflexive and transitive but not symmetric.

Want Us to Email you About Special Offers & Updates?

Q.
Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]