JEE MAIN - Mathematics (2024 - 4th April Evening Shift - No. 18)
Let a relation $$\mathrm{R}$$ on $$\mathrm{N} \times \mathbb{N}$$ 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) $$\mathrm{R}$$ is transitive
Then which one of the following is true?
Explanation
$$\begin{aligned} & \left(x_1, y_1\right) R\left(x_2, y_2\right) \\ & \text { If } x_1 \leq x_2 \text { or } y_1 \leq y_2 \end{aligned}$$
For reflexive;
$$\begin{aligned} & \left(x_1, y_1\right) R\left(x_1, y_1\right) \\ & \Rightarrow x_1 \leq x_1 \text { or } y_1 \leq y_1 \end{aligned}$$
So, $$R$$ is reflexive
For symmetric
When $$\left(x_1, y_1\right) R\left(x_2, y_2\right)$$
$$\Rightarrow x_1 \leq x_2 \text { or } y_1 \leq y_2$$
For $$\left(x_2, y_2\right) R\left(x_1, y_1\right)$$
$$\Rightarrow x_2 \leq x_1 \text { or } y_2 \leq y_1$$
Not true for $$(1,2)$$ and $$(3,4)$$
For transitive
Take pairs as $$(3,9),(4,6),(2,7)$$
$$(3,9) R(4,6)$$
as $$4 \geq 3$$
$$(4,6) R(2,7)$$
As $$7 \geq 6$$
But $$(3,9) R(2,7)$$
As neither $$2 \geq 3$$ nor $$7 \geq 9$$
So not transitive
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