Suppose there is a relation between the positive numbers x and y given by x y if and only if x ...........

Mathematics (NDA) Relations and Functions

Suppose, there is a relation * between the positive numbers x and y given by x * y if and only if x $\leq$ y² . Then which one of the following is correct?

* is reflexive but not transitive and symmetric

* is transitive but not reflexive and symmetric

* is symmetric and reflexive but not transitive

* is symmetric but not reflexive and transitive

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Explanation:

(i) Reflexive
Given, xRy => x is less than y².
$\therefore$ xRx => x is less than x².
Which is not true.
Hence, R is not reflexive.
(ii) Symmetric
xRy => x is less than y².
yRx => y is less than x² is not true because
1R2 => 1 is less than 2².
2R1 => 2 is less than 1².
It is not symmetric.
Hence, option (b) is correct.