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

The Correct Option is B

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.