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

The problem provides a set \( A = \{-3, -2, -1, 0, 1, 2, 3\} \) and a relation R on A defined by the condition \( xRy \) if and only if \( 0 \le x^2 + 2y \le 4 \). We are asked to find the sum \( l + m \), where \( l \) is the number of elements in R, and \( m \) is the minimum number of elements to add to R to make it a reflexive relation.

Concept Used:

1. Relation: A relation R on a set A is a subset of the Cartesian product \( A \times A \). An ordered pair \( (x, y) \) is in R if it satisfies the given condition.

2. Cardinality of a Relation: The number of elements in a relation R, denoted by \( l \), is the total count of ordered pairs \( (x, y) \) that satisfy the condition for the relation.

3. Reflexive Relation: A relation R on a set A is reflexive if for every element \( a \in A \), the ordered pair \( (a, a) \) is in R. The minimum number of elements, \( m \), required to make R reflexive is the count of pairs \( (a, a) \) that are not already in R.

Step-by-Step Solution:

Step 1: Determine the number of elements in R, which is \( l \).

The condition for \( (x, y) \in R \) is \( 0 \le x^2 + 2y \le 4 \), where \( x, y \in A \). We can rearrange this inequality to find the possible values of y for each x:

\[ -x^2 \le 2y \le 4 - x^2 \] \[ -\frac{x^2}{2} \le y \le \frac{4 - x^2}{2} \]

We will now test each value of \( x \in A \) to find the corresponding integer values of \( y \in A \).

• For \( x = -3 \) or \( x = 3 \): \( x^2 = 9 \). The inequality for y is \( -\frac{9}{2} \le y \le \frac{4 - 9}{2} \), which simplifies to \( -4.5 \le y \le -2.5 \). The only integer in A that satisfies this is \( y = -3 \). This gives us the pairs \( (-3, -3) \) and \( (3, -3) \). (2 elements)
• For \( x = -2 \) or \( x = 2 \): \( x^2 = 4 \). The inequality for y is \( -\frac{4}{2} \le y \le \frac{4 - 4}{2} \), which simplifies to \( -2 \le y \le 0 \). The integers in A that satisfy this are \( y = -2, -1, 0 \). This gives the pairs \( (-2, -2), (-2, -1), (-2, 0) \) and \( (2, -2), (2, -1), (2, 0) \). (6 elements)
• For \( x = -1 \) or \( x = 1 \): \( x^2 = 1 \). The inequality for y is \( -\frac{1}{2} \le y \le \frac{4 - 1}{2} \), which simplifies to \( -0.5 \le y \le 1.5 \). The integers in A that satisfy this are \( y = 0, 1 \). This gives the pairs \( (-1, 0), (-1, 1) \) and \( (1, 0), (1, 1) \). (4 elements)
• For \( x = 0 \): \( x^2 = 0 \). The inequality for y is \( -\frac{0}{2} \le y \le \frac{4 - 0}{2} \), which simplifies to \( 0 \le y \le 2 \). The integers in A that satisfy this are \( y = 0, 1, 2 \). This gives the pairs \( (0, 0), (0, 1), (0, 2) \). (3 elements)

The total number of elements in R is \( l = 2 + 6 + 4 + 3 = 15 \).

Step 2: Determine the number of elements to add to make R reflexive, which is \( m \).

For R to be reflexive, it must contain all pairs \( (a, a) \) for every \( a \in A \). We check if \( (a, a) \in R \) by testing if \( 0 \le a^2 + 2a \le 4 \).

• For \( a = -3 \): \( (-3)^2 + 2(-3) = 9 - 6 = 3 \). \( 0 \le 3 \le 4 \). So, \( (-3, -3) \in R \).
• For \( a = -2 \): \( (-2)^2 + 2(-2) = 4 - 4 = 0 \). \( 0 \le 0 \le 4 \). So, \( (-2, -2) \in R \).
• For \( a = -1 \): \( (-1)^2 + 2(-1) = 1 - 2 = -1 \). The condition \( 0 \le -1 \) is false. So, \( (-1, -1) \notin R \).
• For \( a = 0 \): \( 0^2 + 2(0) = 0 \). \( 0 \le 0 \le 4 \). So, \( (0, 0) \in R \).
• For \( a = 1 \): \( 1^2 + 2(1) = 1 + 2 = 3 \). \( 0 \le 3 \le 4 \). So, \( (1, 1) \in R \).
• For \( a = 2 \): \( 2^2 + 2(2) = 4 + 4 = 8 \). The condition \( 8 \le 4 \) is false. So, \( (2, 2) \notin R \).
• For \( a = 3 \): \( 3^2 + 2(3) = 9 + 6 = 15 \). The condition \( 15 \le 4 \) is false. So, \( (3, 3) \notin R \).

The pairs that need to be added to make R reflexive are \( (-1, -1) \), \( (2, 2) \), and \( (3, 3) \). Therefore, the number of elements to be added is \( m = 3 \).

Final Computation & Result:

We are asked to find the value of \( l + m \).

\[ l = 15 \] \[ m = 3 \] \[ l + m = 15 + 3 = 18 \]

The value of \( l + m \) is 18.

Answer 2

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)