Let a relation R on N × N be defined as: (x_1, y_1 ) R (x_2, y_2 ) if and only if x_1 ≤ x_2 or y_1 ≤ y_2 . Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true?

Correct answer: (D) Only (I) is correct.. All ( (x_1 y_1 ), (x_1, y_1 ) ) are in R where x_1, y_1 ∈ N ∴ R is reflexive ((1,1),(2,3)) ∈ R but ((2,3),(1,1)) ∉ R ∴ R is not symmetric ((2,4),(3,3)) ∈ R and ((3,3),(1,3)) ∈ R but ((2,4) , (1,3)) ∉ R ∴ R is not transitive

Q47 / 90 · JEE Main 2024 (04 Apr Shift 2) · Mathematics

Let a relation $\mathrm{R}$ on $\mathrm{N} \times N$ be defined as: $\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements: (I) $\mathrm{R}$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

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

Solution

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

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

Solution

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