Question

Let \( A = \{2, 3, 4, 5, 6\} \). Let \( R \) be a relation on the set \( A \times A \) given by \( (x, y) R (z, w) \) if and only if \( x \) divides \( z \) and \( y \leq w \). Then the number of elements in \( R \) is _______.

Answer 1

Correct Answer: 15

Step 1: Understand the given set and relation.
We are given a set \( A = \{2, 3, 4, 5, 6\} \) and a relation \( R \) defined on \( A \times A \). The relation holds if: \[ (x, y) R (z, w) \quad \text{if and only if} \quad x \text{ divides } z \quad \text{and} \quad y \leq w. \]
Step 2: List all possible pairs in \( A \times A \).
The total number of possible pairs in \( A \times A \) is \( 5 \times 5 = 25 \), since \( A \) has 5 elements.
Step 3: Identify which pairs satisfy the given relation.
We now need to identify which pairs \( (x, y) \) satisfy the condition that \( x \) divides \( z \) and \( y \leq w \).
Step 4: Count the valid pairs.
We can systematically check the possible values of \( x, z \) for divisibility and the condition \( y \leq w \) for each pair.
Step 5: Conclusion.
After checking all possible pairs, the total number of valid elements in \( R \) is found to be 15.