Question

Let $ A = \{-2, -1, 0, 1, 2, 3\} $. Let $ R $ be a relation on $ A $ defined by $ (x, y) \in R $ if and only if $ |x| \le |y| $. Let $ m $ be the number of reflexive elements in $ R $ 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. 13
• B. 12
• C. 11
• D. 14

Hint:

A relation is reflexive if \( (x, x) \) is in the relation for all elements \( x \) in the set. A relation is symmetric if whenever \( (x, y) \) is in the relation, \( (y, x) \) is also in the relation. To make a relation reflexive, add all missing pairs of the form \( (x, x) \). To make a relation symmetric, for every pair \( (x, y) \) in the relation, if \( (y, x) \) is not already present, add it.

Answer 1

The Correct Option is A

Let the set \( A = \{-2, -1, 0, 1, 2, 3\} \) 
Let the relation \( R = \{(-2,1), (-1,1), (0,1), (1,1), (2,2), (3,3)\} \) 
\[\begin{align*} \lambda &= 6 \\m &= 3 \\n &= 3 \\\lambda + m + n &= 6 + 3 + 3 = 12 \end{align*}\]