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

To solve this problem, we need to check the symmetry of the relation \( R \) on the set \( M = \{ 1, 2, 3, \dots, 16 \} \) defined by \( xRy \) if and only if \( 4y = 5x - 3 \). A relation is symmetric if for every pair \( (x, y) \) in the relation, the pair \( (y, x) \) is also in the relation.

Let's find the ordered pairs \( (x, y) \) that satisfy the given relation:

The equation for the relation is:

\(4y = 5x - 3 \Rightarrow y = \frac{5x - 3}{4}\)

For \( y \) to be an integer, \( 5x - 3 \) must be divisible by 4. We check for which integer values of \( x \) (from 1 to 16) this happens:

• For \( x = 1 \), \( 5 \times 1 - 3 = 2 \), not divisible by 4.
• For \( x = 2 \), \( 5 \times 2 - 3 = 7 \), not divisible by 4.
• For \( x = 3 \), \( 5 \times 3 - 3 = 12 \), divisible by 4. Hence, \( y = \frac{12}{4} = 3 \). So, the pair \((3, 3)\) is in \( R \).
• For \( x = 4 \), \( 5 \times 4 - 3 = 17 \), not divisible by 4.
• For \( x = 5 \), \( 5 \times 5 - 3 = 22 \), not divisible by 4.
• For \( x = 6 \), \( 5 \times 6 - 3 = 27 \), not divisible by 4.
• For \( x = 7 \), \( 5 \times 7 - 3 = 32 \), divisible by 4. Hence, \( y = \frac{32}{4} = 8 \). So, the pair \((7, 8)\) is in \( R \).
• For \( x = 8 \), \( 5 \times 8 - 3 = 37 \), not divisible by 4.
• For \( x = 9 \), \( 5 \times 9 - 3 = 42 \), divisible by 4. Hence, \( y = \frac{42}{4} = 10.5 \), not an integer.
• For \( x = 10 \), \( 5 \times 10 - 3 = 47 \), not divisible by 4.
• For \( x = 11 \), \( 5 \times 11 - 3 = 52 \), divisible by 4. Hence, \( y = \frac{52}{4} = 13 \). So, the pair \((11, 13)\) is in \( R \).
• For \( x = 12 \), \( 5 \times 12 - 3 = 57 \), not divisible by 4.
• For \( x = 13 \), \( 5 \times 13 - 3 = 62 \), not divisible by 4.
• For \( x = 14 \), \( 5 \times 14 - 3 = 67 \), not divisible by 4.
• For \( x = 15 \), \( 5 \times 15 - 3 = 72 \), divisible by 4. Hence, \( y = \frac{72}{4} = 18 \), but \( 18 \notin M \).
• For \( x = 16 \), \( 5 \times 16 - 3 = 77 \), not divisible by 4.

Thus, the pairs in the relation are \((3, 3)\), \((7, 8)\), and \((11, 13)\).

Next, to check for symmetry, we need to ensure that if \((x, y)\) is in the relation then \((y, x)\) should also be present in \( R \).

• The pair \((3, 3)\) is symmetric by itself.
• \((7, 8)\) must check if \((8, 7)\) is a part of \( R \). To check: \( 4 \times 7 = 5 \times 8 - 3 \) does not satisfy the condition.
• \((11, 13)\) must check if \((13, 11)\) is a part of \( R \). To check: \( 4 \times 11 = 5 \times 13 - 3 \) does not satisfy the condition.

To make the relation symmetric, we must add the pairs \((8, 7)\) and \((13, 11)\). Thus, 2 more elements need to be added to make the relation symmetric.

Therefore, the number of elements required to be added in \( R \) to make it symmetric is \(2\).