Question

Let \( M = \{ 1, 2, 3, \dots, 16 \} \) and \( R \) be a relation on \( M \) defined by \( xRy \) if and only if \( 4y = 5x - 3 \). Then, the number of elements required to be added in \( R \) to make it symmetric is:

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

To make a relation symmetric, check that for every pair \( (x, y) \) in the relation, the reverse pair \( (y, x) \) must also be included.

Answer 1

The Correct Option is A

Step 1: Understanding the relation.
The relation \( R \) is defined as \( xRy \) if and only if \( 4y = 5x - 3 \). For the relation to be symmetric, if \( xRy \) is true, then \( yRx \) must also be true. Step 2: Finding pairs for symmetry.
For symmetry, we need to ensure that the relation \( 4y = 5x - 3 \) also holds for \( 4x = 5y - 3 \). Solving the system of equations for pairs of \( x \) and \( y \), we find the pairs that violate symmetry. Step 3: Conclusion.
After analyzing, we find that 2 additional elements need to be added to make the relation symmetric. Final Answer: \[ \boxed{2} \]