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 first need to find all the elements (ordered pairs) in R. Then, we need to determine how many more ordered pairs must be added to R to satisfy the property of symmetry.

Step 2: Finding the Elements of Relation R:
The relation is defined by the equation \(4y = 5x - 3\), where \(x, y \in \{1, 2, ..., 16\}\).
We can write this as \(y = \frac{5x - 3}{4}\). For y to be an integer, \(5x - 3\) must be divisible by 4.
We can check this condition using modular arithmetic: \(5x - 3 \equiv 0 \pmod{4}\).
Since \(5 \equiv 1 \pmod{4}\) and \(-3 \equiv 1 \pmod{4}\), the condition becomes \(x + 1 \equiv 0 \pmod{4}\), or \(x \equiv -1 \pmod{4}\), which is \(x \equiv 3 \pmod{4}\).
This means x must be of the form \(4k+3\) for some integer k.
We test values of x in M = \{1, 2, ..., 16\} that satisfy this condition:
If \(x = 3\): \(y = \frac{5(3) - 3}{4} = \frac{12}{4} = 3\). Since \(y=3 \in M\), the pair (3, 3) is in R.
If \(x = 7\): \(y = \frac{5(7) - 3}{4} = \frac{32}{4} = 8\). Since \(y=8 \in M\), the pair (7, 8) is in R.
If \(x = 11\): \(y = \frac{5(11) - 3}{4} = \frac{52}{4} = 13\). Since \(y=13 \in M\), the pair (11, 13) is in R.
If \(x = 15\): \(y = \frac{5(15) - 3}{4} = \frac{72}{4} = 18\). Since \(y=18 \notin M\), this pair is not in R. So, the relation R is \(R = \{(3, 3), (7, 8), (11, 13)\}\).

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