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STD 12 - 1. relation and function — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 1. relation and function1 Mark
MCQ
Let a relation $R$ on $\mathbb{N} \times \mathbb{N}$ be defined as : $\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ if and only if $\mathrm{x}_1 \leq \mathrm{x}_2$ or $\mathrm{y}_1 \leq \mathrm{y}_2$

Consider the two statements :

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

($II$) $\mathrm{R}$ is transitive

Then which one of the following is true?

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

Answer

Correct option: B.
 Only ($I$) is correct.
b
All $\left(\left(\mathrm{x}_1 \mathrm{y}_1\right),\left(\mathrm{x}_1, \mathrm{y}_1\right)\right)$ are in $\mathrm{R}$ where

$\mathrm{x}_1, \mathrm{y}_1 \in \mathrm{N} \therefore \mathrm{R}$ is reflexive

$((1,1),(2,3)) \in \mathrm{R}$ but $((2,3),(1,1)) \notin \mathrm{R}$

$\therefore \mathrm{R}$ is not symmetric

$((2,4),(3,3)) \in \mathrm{R}$ and $((3,3),(1,3)) \in \mathrm{R}$ but $((2,4)$,

$(1,3)) \notin \mathrm{R}$

$\therefore \mathrm{R}$ is not transitive

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