Question

Let $ A = \{-3, -2, -1, 0, 1, 2, 3\} $ and $ R $ be a relation on $ A $ defined by $ xRy $ if and only if $ 2x - y \in \{0, 1\} $. Let $ l $ be the number of elements in $ R $. Let $ m $ 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. 18
• B. 17
• C. 15
• D. 16

Hint:

Check for reflexivity and symmetry when working with relations on a set to ensure completeness.

Answer 1

The Correct Option is B

The given relation is defined by \( 2x - y \in \{0, 1\} \). By checking all possible pairs, we find the following: \[ R = \{(0, 0), (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3)\} \] The number of elements in \( R \) is 7. For reflexivity, we need to add the following elements: \( (0, 0), (1, 1), (2, 2), (-1, -1), (-2, -2), (3, 3) \), which means 5 elements need to be added. For symmetry, we need to add the pairs: \[ (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3) \]
Thus, \( l + m + n = 17 \).
Thus, the correct answer is \( 17 \).

Answer 2

Step 1: Understand the relation.  
Given \( A = \{-3, -2, -1, 0, 1, 2, 3\} \), the relation \( R \) is defined as: 
\[ xRy \text{ if and only if } 2x - y \in \{0, 1\} \] 
This means that for each \( x \) and \( y \), the condition \( 2x - y = 0 \) or \( 2x - y = 1 \) must hold for \( xRy \). 
Step 2: Find the elements in the relation. 
We can now check the values of \( x \) and \( y \) to find the relation elements \( R \). 
- For \( x = -3 \), no \( y \) satisfies the condition \( 2x - y = 0 \) or \( 2x - y = 1 \). 
- For \( x = -2 \), \( 2(-2) - (-2) = -4 + 2 = -2 \) does not satisfy the condition. 
- Similarly, we proceed for all values of \( x \) and identify the relation pairs. 
Step 3: Reflexive and symmetric relations. 
- Reflexive: A relation is reflexive if for every \( x \in A \), \( xRx \) is included. Therefore, the diagonal elements must be included in the relation. 
- Symmetric: A relation is symmetric if \( xRy \) implies \( yRx \). We will ensure all pairs are symmetric. 
Step 4: Finding the minimum number of elements required to make the relation reflexive and symmetric. 
The number of elements required to make the relation reflexive and symmetric is the sum of the minimum elements needed for each condition. 
Step 5: Conclusion. 
After calculating \( l \) and \( m \), the value of \( l + m + n \) is \( 17 \). 
Final Answer: 
\[ \boxed{17}. \]