Question

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

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

Answer 1

Answer: (B)

Only (I) is correct.

Answer 2

To verify the properties of $R$, consider all $(x_1, y_1), (x_2, y_2) \in R$ where $x_1, y_1 \in \mathbb{N}$.

1. $R$ is reflexive: For all $(x_1, y_1) \in \mathbb{N} \times \mathbb{N}$, $x_1 \leq x_1 \, \text{or} \, y_1 \leq y_1$ is always true.
   Hence, $R$ is reflexive.
2. $R$ is not symmetric: For example, consider $(1, 2) R (2, 3)$ because $1 \leq 2$. However, $(2, 3) \notin R(1, 2)$ because neither $2 \leq 1$ nor $3 \leq 2$. Hence, $R$ is not symmetric.
3. $R$ is not transitive: For example, consider $(2, 4)R(3, 3)$ and $(3, 3)R(1, 3)$. However, $(2, 4) \notin R(1, 3)$, so $R$ is not transitive.

Thus, only statement (I) is correct.