Question

Let \( A = \{-2,-1,0,1,2,3,4\} \) and \( R \) be a relation such that \[ R=\{(x,y): 2x+y \le -2,\ x\in A,\ y\in A\}. \] Let
\( l \) = number of elements in \( R \),
\( m \) = minimum number of elements to be added to \( R \) to make it reflexive,
\( n \) = minimum number of elements to be added to \( R \) to make it symmetric. Then \( (l+m+n) \) is:

• A. \(10\)
• B. \(17\)
• C. \(11\)
• D. \(14\)

Hint:

For symmetric relations, always check whether the reverse ordered pair already exists before counting additions.

Answer 1

The Correct Option is B

Concept:

A relation is reflexive if \( (a,a)\in R \) for all \( a\in A \).
A relation is symmetric if \( (x,y)\in R \Rightarrow (y,x)\in R \).
Step 1: Find the number of elements in \( R \). We check \( 2x+y \le -2 \) for each \( x\in A \).

\[ l = 5+3+1 = 9 \]
Step 2: Find \( m \) to make \( R \) reflexive. Reflexive condition: \( (a,a)\in R \Rightarrow 3a \le -2 \) This holds only for: \[ a=-2,-1 \] Already present: \( (-2,-2), (-1,-1) \) Total elements in \( A =7 \) \[ m = 7-2 = 5 \]
Step 3: Find \( n \) to make \( R \) symmetric. Check pairs whose reverse is missing: \[ (-2,1) \Rightarrow (1,-2) \notin R \] \[ (-2,2) \Rightarrow (2,-2) \notin R \] \[ (-1,0) \Rightarrow (0,-1) \notin R \] Thus, \[ n=3 \]
Step 4: Final calculation: \[ l+m+n = 9+5+3 = 17 \]