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 relation \( 2x - y \in \{0, 1\} \) is defined. The relation \( R \) consists of the following pairs: \( R = \{(0, 0), (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3)\} \). The set \( R \) has 7 elements. To ensure reflexivity, the elements \( (0, 0), (1, 1), (2, 2), (-1, -1), (-2, -2), (3, 3) \) must be included, requiring the addition of 5 new elements. For symmetry, the pairs \( (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3) \) must be added.
Therefore, \( l + m + n = 17 \).
The final answer is \( 17 \).