Question

Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). A relation \( R \) is defined such that \( xRy \) if \( y = \max(x, 1) \). The number of elements required to make it reflexive is \( l \), the number of elements required to make it symmetric is \( m \), and the number of elements in the relation \( R \) is \( n \). Then the value of \( l + m + n \) is equal to:

• A. 7
• B. 8
• C. 15
• D. 10

Hint:

To make a relation reflexive, ensure that every element is related to itself. To make it symmetric, ensure that for every \( (x, y) \), the pair \( (y, x) \) is also included.

Answer 1

The Correct Option is C

Step-by-Step Solution

Step 1: Understand the Set and Relation

The set \( A = \{-3, -2, -1, 0, 1, 2, 3\} \) has 7 elements, so \( |A| = 7 \). The relation \( R \) is defined as \( y = \max(x, 1) \). We need to find all pairs \((x, y)\) in \( R \).

Step 2: Find Elements of \( R \)

Let’s compute \( y = \max(x, 1) \) for each \( x \in A \):

• For \( x = -3 \), \( y = \max(-3, 1) = 1 \), so pair is \((-3, 1)\).
• For \( x = -2 \), \( y = \max(-2, 1) = 1 \), so pair is \((-2, 1)\).
• For \( x = -1 \), \( y = \max(-1, 1) = 1 \), so pair is \((-1, 1)\).
• For \( x = 0 \), \( y = \max(0, 1) = 1 \), so pair is \((0, 1)\).
• For \( x = 1 \), \( y = \max(1, 1) = 1 \), so pair is \((1, 1)\).
• For \( x = 2 \), \( y = \max(2, 1) = 2 \), so pair is \((2, 2)\).
• For \( x = 3 \), \( y = \max(3, 1) = 3 \), so pair is \((3, 3)\).

So, \( R = \{(-3, 1), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\} \). The number of pairs in \( R \) is 7, so \( n = 7 \).

Step 3: Find \( l \) (Elements to Make \( R \) Reflexive)

A relation is reflexive if for every \( x \in A \), the pair \((x, x) \in R \). The required pairs are \((-3, -3), (-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)\).

Check which are in \( R \):

• \((-3, -3)\), \((-2, -2)\), \((-1, -1)\), \((0, 0)\): Not in \( R \).
• \((1, 1)\), \((2, 2)\), \((3, 3)\): Already in \( R \).

We need to add 4 pairs: \((-3, -3), (-2, -2), (-1, -1), (0, 0)\). So, \( l = 4 \).

Step 4: Find \( m \) (Elements to Make \( R \) Symmetric)

A relation is symmetric if for every \((x, y) \in R \), the pair \((y, x) \in R \). Check each pair:

• \((-3, 1)\): Need \((1, -3)\). Not in \( R \). Add \((1, -3)\).
• \((-2, 1)\): Need \((1, -2)\). Not in \( R \). Add \((1, -2)\).
• \((-1, 1)\): Need \((1, -1)\). Not in \( R \). Add \((1, -1)\).
• \((0, 1)\): Need \((1, 0)\). Not in \( R \). Add \((1, 0)\).
• \((1, 1)\), \((2, 2)\), \((3, 3)\): Already symmetric.

We need to add 4 pairs: \((1, -3), (1, -2), (1, -1), (1, 0)\). So, \( m = 4 \).

Step 5: Compute \( l + m + n \)

We have \( l = 4 \), \( m = 4 \), \( n = 7 \).

\( l + m + n = 4 + 4 + 7 = 15 \).

Final Answer: \( l + m + n = 15 \)

Conclusion

This step-by-step solution shows how to determine the elements needed to make a relation reflexive and symmetric, a key concept in Class 12 math and JEE Main EXam Understanding relations is crucial for discrete math and CBSE exams. Practice more problems like this to master the topic!