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

Process Summary

Stage 1: Define Set and Relation

The set is \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). The size of set A is \( |A| = 7 \). The relation R is defined by \( y = \max(x, 1) \). We aim to identify all ordered pairs \((x, y)\) belonging to R.

Stage 2: Enumerate Elements of \( R \)

We calculate \( y = \max(x, 1) \) for each \( x \) in \( A \):

• For \( x = -3 \), \( y = \max(-3, 1) = 1 \). Pair: \((-3, 1)\).
• For \( x = -2 \), \( y = \max(-2, 1) = 1 \). Pair: \((-2, 1)\).
• For \( x = -1 \), \( y = \max(-1, 1) = 1 \). Pair: \((-1, 1)\).
• For \( x = 0 \), \( y = \max(0, 1) = 1 \). Pair: \((0, 1)\).
• For \( x = 1 \), \( y = \max(1, 1) = 1 \). Pair: \((1, 1)\).
• For \( x = 2 \), \( y = \max(2, 1) = 2 \). Pair: \((2, 2)\).
• For \( x = 3 \), \( y = \max(3, 1) = 3 \). Pair: \((3, 3)\).

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

Stage 3: Determine \( l \) (Pairs for Reflexivity)

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

We check which of these are present in R:

• \((-3, -3)\), \((-2, -2)\), \((-1, -1)\), \((0, 0)\) are not in \( R \).
• \((1, 1)\), \((2, 2)\), \((3, 3)\) are already in \( R \).

To achieve reflexivity, we must add the 4 pairs: \((-3, -3), (-2, -2), (-1, -1), (0, 0)\). Therefore, \( l = 4 \).

Stage 4: Determine \( m \) (Pairs for Symmetry)

A relation is symmetric if whenever \((x, y) \in R\), then \((y, x) \in R\). We examine each pair in R:

• For \((-3, 1) \in R\), we require \((1, -3)\). This is not in \( R \). Add \((1, -3)\).
• For \((-2, 1) \in R\), we require \((1, -2)\). This is not in \( R \). Add \((1, -2)\).
• For \((-1, 1) \in R\), we require \((1, -1)\). This is not in \( R \). Add \((1, -1)\).
• For \((0, 1) \in R\), we require \((1, 0)\). This is not in \( R \). Add \((1, 0)\).
• Pairs \((1, 1)\), \((2, 2)\), \((3, 3)\) are self-symmetric.

To achieve symmetry, we must add the 4 pairs: \((1, -3), (1, -2), (1, -1), (1, 0)\). Therefore, \( m = 4 \).

Stage 5: Calculate \( l + m + n \)

Given \( l = 4 \), \( m = 4 \), and \( n = 7 \).

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

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

Summary

This analysis details the process of finding pairs to make a relation reflexive and symmetric. This is a fundamental concept in higher mathematics, particularly relevant for advanced coursework and competitive examinations. Continued practice with similar problems will solidify understanding of relations and their properties.