Question

Let $ A = \{-3, -2, -1, 0, 1, 2, 3\} $ and $ R $ be a relation on $ A $ defined by $ xRy $ if and only if $ 2x - y \in \{0, 1\} $. Let $ l $ be the number of elements in $ R $. Let $ m $ and $ n $ be the minimum number of elements required to be added in $ R $ to make it reflexive and symmetric relations, respectively. Then $ l + m + n $ is equal to:

• A. 18
• B. 17
• C. 15
• D. 16

Hint:

Check for reflexivity and symmetry when working with relations on a set to ensure completeness.

Answer 1

The Correct Option is B

The given relation is defined by \( 2x - y \in \{0, 1\} \). By checking all possible pairs, we find the following: \[ R = \{(0, 0), (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3)\} \] The number of elements in \( R \) is 7. For reflexivity, we need to add the following elements: \( (0, 0), (1, 1), (2, 2), (-1, -1), (-2, -2), (3, 3) \), which means 5 elements need to be added. For symmetry, we need to add the pairs: \[ (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3) \]
Thus, \( l + m + n = 17 \).
Thus, the correct answer is \( 17 \).