Verified Solution • 2025–26 Session moderate

QJEEM24Mabc95: JEE Main 2024 Mathematics - Class 11 JEEM24Mabc95 Solution: Let a relation on be defined as: if and only if or . Consider…

Let a relation R\mathrm{R} on N×N\mathrm{N} \times \mathbb{N} be defined as: (x1,y1)R(x2,y2)\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right) if and only if x1≤x2x_1 \leq x_2 or y1≤y2y_1 \leq y_2 . Consider the two statements:

(I) R\mathrm{R} is reflexive but not symmetric.

(II) R\mathrm{R} is transitive

Then which one of the following is true?

Multiple Choice Options

  • A Only (II) is correct.
  • B Both (I) and (II) are correct.
  • C Neither (I) nor (II) is correct.
  • D Only (I) is correct.

Step-by-Step Verified Solution

amp;(x1,y1)R(x2,y2)amp; If x1≤x2 or y1≤y2\begin{aligned}& \left(x_1, y_1\right) R\left(x_2, y_2\right) \\& \text { If } x_1 \leq x_2 \text { or } y_1 \leq y_2\end{aligned}

For reflexive;

amp;(x1,y1)R(x1,y1)amp;⇒x1≤x1 or y1≤y1\begin{aligned}& \left(x_1, y_1\right) R\left(x_1, y_1\right) \\& \Rightarrow x_1 \leq x_1 \text { or } y_1 \leq y_1\end{aligned}

So, RR is reflexive

For symmetric

When (x1,y1)R(x2,y2)\left(x_1, y_1\right) R\left(x_2, y_2\right)

⇒x1≤x2 or y1≤y2\Rightarrow x_1 \leq x_2 \text { or } y_1 \leq y_2

For (x2,y2)R(x1,y1)\left(x_2, y_2\right) R\left(x_1, y_1\right)

⇒x2≤x1 or y2≤y1\Rightarrow x_2 \leq x_1 \text { or } y_2 \leq y_1

Not true for (1,2)(1,2) and (3,4)(3,4)

For transitive

Take pairs as (3,9),(4,6),(2,7)(3,9),(4,6),(2,7)

(3,9)R(4,6)(3,9) R(4,6)

as 4≥34 \geq 3

(4,6)R(2,7)(4,6) R(2,7)

As 7≥67 \geq 6

But (3,9)R(2,7)(3,9) R(2,7)

As neither 2≥32 \geq 3 nor 7≥97 \geq 9

So not transitive

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