Q152 / 270 · NDA 2016 (Phase 1) · Mathematics

Suppose there is a relation $*$ between the positive numbers x and y given by $\mathrm{x} * \mathrm{y}$ if and only if $\mathrm{x} \leq \mathrm{y}^2$. Then which one of the following is correct?

  • Correct answer
    A
    *is reflexive but not transitive and symmetric
  • B
    *is transitive but not reflexive and symmetric
  • C
    *is symmetric and reflexive but not transitive
  • D
    *is symmetric and but not reflexive and transitive

Solution

$x$ and $y$ are positive numbers. $x \leq y^2$ Reflexive- $x \leq x^2 \forall$ positive numbers. Hence relation is reflexive. Transitive- $x \leq y^2 \quad y \leq z^2$ Let $x=5, y=3, z=2$ $5 \leq(3)^2 \quad(3) \leq(2)^2$ but $5 \not(2)^2$ Hence, $x \leq y^2$ $y \leq z^2$ but $x \not \leq z^2$ Thus relation is not transitive. Symmetric- $1 \leq(2)^2$ while $2 \not(1)^2$ Hence relation is not symmetric. Thus $x \leq y^2 \quad \forall$ positive numbers is reflexive, but not transitive and symmetric.

Hence, the correct answer is option: 'A'.

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