Question

Let the relation \( R \) on the set \( M = \{1, 2, 3, \ldots, 16\} \) be given by \[ R = \{(x, y) : 4y = 5x - 3,\; x, y \in M\}. \] Then the minimum number of elements required to be added in \( R \), in order to make the relation symmetric, is equal to

• A. 3
• B. 4
• C. 2
• D. 1

Hint:

To make a relation symmetric, ensure that for every ordered pair \( (x,y) \), the reverse pair \( (y,x) \) is also included. Count only the missing reverse pairs.

Answer 1

The Correct Option is B

The given question involves determining the number of elements to be added for a relation \( R \) to be symmetric. The relation \( R \) on the set \( M = \{1, 2, 3, \ldots, 16\} \) is defined as:

\(R = \{(x, y) : 4y = 5x - 3, x, y \in M\}\).

In mathematical terms, a relation is symmetric if for every \((a, b) \in R\), there exists \((b, a) \in R\).

Let's find the elements of \( R \) that satisfy the equation:

• \(4y = 5x - 3\) implies \(y = \frac{5x - 3}{4}\).

Since \( y \) must be an integer, \(5x - 3\) should be divisible by 4. Let's find the pairs \((x, y)\):

x	y	(x, y)
1	1	(1, 1)
3	3	(3, 3)
4	5	(4, 5)
7	8	(7, 8)
11	13	(11, 13)
12	15	(12, 15)
16	20	(16, 20)

The pairs \((x, y)\) where both \(x\) and \(y\) are less than or equal to 16 are: (1, 1), (3, 3), (4, 5), (7, 8), (11, 13), and (12, 15).

To make the relation symmetric, for each non-symmetric pair \((x, y)\), we need to check if \((y, x)\) exists. Let's check:

• (1, 1): Symmetric as (1, 1) exists.
• (3, 3): Symmetric as (3, 3) exists.
• (4, 5): Check for (5, 4) in \( R \).
• (7, 8): Check for (8, 7) in \( R \).
• (11, 13): Check for (13, 11) in \( R \).
• (12, 15): Check for (15, 12) in \( R \).

None of the reversed pairs (for the non-trivial cases) already exist in \( R \). Therefore, we need to add these reversed pairs to make the relation symmetric:

• (5, 4), (8, 7), (13, 11), (15, 12)

Thus, we need a minimum of \(4\) additional pairs to make \( R \) symmetric.

Therefore, the correct answer is: 4 elements need to be added.