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

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

Concepts Utilized:

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

2. Cardinality of a Relation: The number of ordered pairs \( (x, y) \) in a relation R, denoted by \( l \), that satisfy the relation's condition.

3. Reflexive Relation: A relation R on a set A is reflexive if \( (a, a) \in R \) for all \( a \in A \). The minimum number of elements, \( m \), to add for reflexivity is the count of \( (a, a) \) pairs not already in R.

Solution Steps:

Step 1: Calculate \( l \), the number of elements in R.

The condition for \( (x, y) \in R \) is \( 0 \le x^2 + 2y \le 4 \) for \( x, y \in A \). Rearranging for y yields:

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

We evaluate this for each \( x \in A \):

• For \( x = \pm 3 \): \( x^2 = 9 \). Inequality: \( -4.5 \le y \le -2.5 \). Only \( y = -3 \in A \). Pairs: \( (-3, -3), (3, -3) \). (2 elements)
• For \( x = \pm 2 \): \( x^2 = 4 \). Inequality: \( -2 \le y \le 0 \). \( y \in \{-2, -1, 0\} \). Pairs: \( (-2, -2), (-2, -1), (-2, 0) \) and \( (2, -2), (2, -1), (2, 0) \). (6 elements)
• For \( x = \pm 1 \): \( x^2 = 1 \). Inequality: \( -0.5 \le y \le 1.5 \). \( y \in \{0, 1\} \). Pairs: \( (-1, 0), (-1, 1) \) and \( (1, 0), (1, 1) \). (4 elements)
• For \( x = 0 \): \( x^2 = 0 \). Inequality: \( 0 \le y \le 2 \). \( y \in \{0, 1, 2\} \). Pairs: \( (0, 0), (0, 1), (0, 2) \). (3 elements)

Total elements in R: \( l = 2 + 6 + 4 + 3 = 15 \).

Step 2: Calculate \( m \), the number of elements needed for R to be reflexive.

A relation is reflexive if \( (a, a) \in R \) for all \( a \in A \). We check \( 0 \le a^2 + 2a \le 4 \):

• \( a = -3 \): \( (-3)^2 + 2(-3) = 3 \). \( 0 \le 3 \le 4 \). \( (-3, -3) \in R \).
• \( a = -2 \): \( (-2)^2 + 2(-2) = 0 \). \( 0 \le 0 \le 4 \). \( (-2, -2) \in R \).
• \( a = -1 \): \( (-1)^2 + 2(-1) = -1 \). \( -1<0 \). \( (-1, -1) otin R \).
• \( a = 0 \): \( 0^2 + 2(0) = 0 \). \( 0 \le 0 \le 4 \). \( (0, 0) \in R \).
• \( a = 1 \): \( 1^2 + 2(1) = 3 \). \( 0 \le 3 \le 4 \). \( (1, 1) \in R \).
• \( a = 2 \): \( 2^2 + 2(2) = 8 \). \( 8>4 \). \( (2, 2) otin R \).
• \( a = 3 \): \( 3^2 + 2(3) = 15 \). \( 15>4 \). \( (3, 3) otin R \).

The pairs to add are \( (-1, -1) \), \( (2, 2) \), and \( (3, 3) \). Thus, \( m = 3 \).

Final Calculation & Result:

We need to compute \( l + m \).

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

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