Let a relation \(R\) on \(\mathbb{N}\times\mathbb{N}\) be defined as: \((x_{1},y_{1}) R (x_{2},y_{2})\) if and only if \(x_{1} \leq x_{2}\) or \(y_{1} \leq y_{2}\). Consider the two statements: (I) \(R\) is reflexive but not symmetric. (II) \(R\) is transitive. Then which one of the following is true?

(A) Neither (I) nor (II) is correct

(C) Both (I) and (II) are correct

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Sets Relations and Functions

Let a relation

R

N \times N

be defined as:

(x_{1}, y_{1}) R (x_{2}, y_{2})

if and only if

x_{1} \leq x_{2}

y_{1} \leq 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?

Answer: (D)

Reflexive: for any

(x, y) \in N \times N

x \leq x

, so

(x, y) R (x, y)

holds —

R

is reflexive.

Symmetric:

(1, 1) R (2, 3)

since

1 \leq 2

, but

(2, 3) R (1, 1)

would need

2 \leq 1

3 \leq 1

, both false —

R

is not symmetric.

Transitive: take

(2, 4) R (3, 3)

(since

2 \leq 3

) and

(3, 3) R (1, 3)

wait — use

(2, 4) R (3, 1)

(since

2 \leq 3

) and

(3, 1) R (1, 3)

(since

1 \leq 3

), but

(2, 4) R (1, 3)

requires

2 \leq 1

4 \leq 3

, both false —

R

is not transitive.

So statement (I) is correct, (II) is not — only (I) is correct.

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Single correct option