Question

Let \(M = \{1, 2, 3, ....., 16\}\), if a relation R defined on set M such that R = \((x, y) : 4y = 5x – 3, x, y (\in) M\). How many elements should be added to R to make it symmetric.

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

To make a relation R symmetric, for every pair \((a, b)\) in R where \(a \neq b\), you must ensure the pair \((b, a)\) is also in R. Count how many such \((b, a)\) pairs are missing.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a set \( M \) and a relation \( R \) on \( M \). We need to find all the elements (ordered pairs) in \( R \), and determine how many more pairs must be added to make \( R \) symmetric. 

Step 2: Finding the Elements of Relation R:
The relation is defined by the equation \( 4y = 5x - 3 \), where \( x, y \in \{1, 2, ..., 16\} \). 
Rewriting, we get \( y = \frac{5x - 3}{4} \). For \( y \) to be an integer, \( 5x - 3 \) must be divisible by 4. We check this using modular arithmetic: \[ 5x - 3 \equiv 0 \pmod{4} \] Since \( 5 \equiv 1 \pmod{4} \) and \( -3 \equiv 1 \pmod{4} \), we have \( x + 1 \equiv 0 \pmod{4} \), or \( x \equiv 3 \pmod{4} \). 
We check values of \( x \in \{1, 2, ..., 16\} \) that satisfy \( x \equiv 3 \pmod{4} \): - For \( x = 3 \), \( y = 3 \) - For \( x = 7 \), \( y = 8 \) - For \( x = 11 \), \( y = 13 \) - For \( x = 15 \), \( y = 18 \) (not in \( M \)) So, \( R = \{(3, 3), (7, 8), (11, 13)\} \). 

Step 3: Making the Relation Symmetric:
A relation \( R \) is symmetric if, for every pair \( (a, b) \in R \), the pair \( (b, a) \) must also be in \( R \). We check: - For \( (3, 3) \), the reverse is \( (3, 3) \), which is already in \( R \). - For \( (7, 8) \), the reverse is \( (8, 7) \), which is not in \( R \), so we add \( (8, 7) \). - For \( (11, 13) \), the reverse is \( (13, 11) \), which is not in \( R \), so we add \( (13, 11) \). 
Step 4: Final Answer:
To make \( R \) symmetric, we need to add the pairs \( (8, 7) \) and \( (13, 11) \). 
The number of pairs to add is 2.