Question

Let A = $\{-3,-2,-1,0,1,2,3\}$. Let R be a relation on A defined by xRy if and only if $ 0 \le x^2 + 2y \le 4 $. Let $ l $ be the number of elements in R and m be the minimum number of elements required to be added in R to make it a reflexive relation. then $ l + m $ is equal to

• A. 19
• B. 20
• C. 17
• D. 18

Hint:

To make a relation reflexive, every element in the set must be related to itself. In other words, for a set A, (a,a) must be in R for all a in A.

Answer 1

The Correct Option is D

Let \( A = \{-3,-2,-1,0,1,2,3\} \) Given: \[ 0 \le x^2 + 2y \le 4 \Rightarrow -2y \le x^2 \le 4 - 2y \] Now for different values of \( y \in A \), find the possible \( x \in A \) satisfying the condition:

• For \( y = -3 \), \( 6 \le x^2 \le 10 \Rightarrow x \in \{-3, 3\} \)
• For \( y = -2 \), \( 4 \le x^2 \le 8 \Rightarrow x \in \{-2, 2\} \)
• For \( y = -1 \), \( 2 \le x^2 \le 6 \Rightarrow x \in \{-2, 2\} \)
• For \( y = 0 \), \( 0 \le x^2 \le 4 \Rightarrow x \in \{-2, -1, 0, 1, 2\} \)
• For \( y = 1 \), \( -2 \le x^2 \le 2 \Rightarrow x \in \{-1, 0, 1\} \)
• For \( y = 2 \), \( -4 \le x^2 \le 0 \Rightarrow x \in \{0\} \)
• For \( y = 3 \), \( -6 \le x^2 \le -2 \Rightarrow \) No such \( x \)

So the relation \( R \) consists of the following ordered pairs: \[ R = \{ (-3,-3), (-3,3), (-2,-2), (-2,2), (-1,-2), (-1,2), (0,-2), (0,-1), (0,0), (0,1), (0,2), (1,-1), (1,0), (1,1), (2,0) \} \] 
Thus, \[ l = |R| = 15 \] To make \( R \) reflexive, we must add the missing self-pairs: 
From set \( A \), reflexive relation requires all \( (a,a) \in A \times A \) 
Already present: \( (0,0) \) 
Missing: \( (-1,-1), (2,2), (3,3) \Rightarrow m = 3 \) 
\[ \therefore l + m = 15 + 3 = 18 \] 
Correct answer: Option (4)