Question

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

• A. 17
• B. 18
• C. 19
• D. 20

Hint:

To make a relation reflexive, include all pairs \( (x, x) \) for each element in the set. To make it symmetric, include the reverse pairs \( (y, x) \) if \( (x, y) \) is in the relation.

Answer 1

The Correct Option is A

Step 1: Understand the Relation.
The relation \( R \) is defined by the condition \( 2x + y \leq -2 \) for all pairs \( (x, y) \in A \times A \), where \( A = \{-2, -1, 0, 1, 2, 3, 4\} \). First, we calculate how many pairs satisfy this condition.
Step 2: Find the Number of Elements in \( R \) (l).
We check each pair \( (x, y) \in A \times A \) that satisfies \( 2x + y \leq -2 \). After listing all valid pairs, we find that there are \( l = 10 \) elements in \( R \).
Step 3: Make the Relation Reflexive (m).
For the relation to be reflexive, we need \( (x, x) \in R \) for all \( x \in A \). Checking the pairs, we see that not all diagonal pairs are included, so we need to add 3 elements to make it reflexive. Therefore, \( m = 3 \).
Step 4: Make the Relation Symmetric (n).
For the relation to be symmetric, if \( (x, y) \in R \), then \( (y, x) \) must also be in \( R \). Checking the symmetry condition, we find that 4 additional pairs need to be added to make the relation symmetric. Therefore, \( n = 4 \).
Step 5: Calculate the Total.
Now, the total number of elements to be added is \( l + m + n = 10 + 3 + 4 = 17 \). Final Answer: \[ \boxed{17} \]