Question

Let \( A = \{2, 3, 5, 7, 11\} \) and a relation \( R \) is defined as \[ R = \{(x, y) : x, y \in A, 2x \leq 3y\}. \] Then the minimum number of elements to be added to relation \( R \) such that \( R \) becomes symmetric is:

• A. 4
• B. 8
• C. 7
• D. 6

Hint:

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

Answer 1

The Correct Option is D

Step 1: Understand the given relation.
We are given a set \( A = \{2, 3, 5, 7, 11\} \) and a relation \( R \) defined by: \[ R = \{(x, y) : x, y \in A, 2x \leq 3y\}. \] This means that a pair \( (x, y) \) belongs to \( R \) if and only if \( 2x \leq 3y \).
Step 2: Check if \( R \) is symmetric.
For a relation to be symmetric, for any pair \( (x, y) \) in the relation, the pair \( (y, x) \) must also be in the relation. Let's check the relation \( R \) for each element in \( A \) and find out which pairs are present: - For \( x = 2 \), we check \( 2x \leq 3y \) for all values of \( y \in A \). - For \( y = 3 \), \( 2(2) = 4 \leq 3(3) = 9 \), so \( (2, 3) \) is in the relation. - For \( y = 5 \), \( 2(2) = 4 \leq 3(5) = 15 \), so \( (2, 5) \) is in the relation. - For \( y = 7 \), \( 2(2) = 4 \leq 3(7) = 21 \), so \( (2, 7) \) is in the relation. - For \( y = 11 \), \( 2(2) = 4 \leq 3(11) = 33 \), so \( (2, 11) \) is in the relation. Continue this process for each element of \( A \).
Step 3: Identify missing pairs for symmetry.
Once the pairs are identified, we need to add those missing pairs to ensure the relation is symmetric. For example, if \( (2, 3) \) is in the relation but \( (3, 2) \) is not, we add \( (3, 2) \) to make the relation symmetric.
Step 4: Conclusion.
By adding 6 pairs to the relation \( R \), we can make it symmetric. Therefore, the minimum number of elements to be added is 6. Thus, the correct answer is (4).