Question

Let \( R \) be a relation defined by the teacher to plan the seating arrangement of students in pairs, where \( R = \{(x, y) : x, y \text{ are Roll Numbers of students such that } y = 3x \} \). List the elements of \( R \). Is the relation \( R \) reflexive, symmetric, and transitive? Justify your answer.

Hint:

For a relation to be reflexive, each element must relate to itself. For symmetry, reverse pairs must also be in the relation. For transitivity, follow the chain of relationships.

Answer 1

The relation \( R \) comprises pairs of student roll numbers where the second roll number is three times the first. If \( x \) is a student's roll number, then \( y = 3x \) is the roll number of another related student. Given student roll numbers \( 1, 2, 3, 4, 5, 6, \dots, 10 \), the relation \( R \) is defined as: \[ R = \{ (1, 3), (2, 6), (3, 9) \} \]
We now assess if \( R \) is reflexive, symmetric, and transitive: 
1. Reflexive: A relation is reflexive if all elements are related to themselves, meaning \( (x, x) \in R \) for all \( x \). 
As \( y = 3x \), the condition \( y = x \) cannot be satisfied unless \( x = 0 \), which is not a typical roll number. Therefore, \( R \) is not reflexive. 
2. Symmetric: A relation is symmetric if \( (x, y) \in R \) implies \( (y, x) \in R \). Given \( y = 3x \), for \( (y, x) \) to be in \( R \), \( x \) would need to equal \( 3y \). Substituting \( y = 3x \), we get \( x = 3(3x) = 9x \), which is only true for \( x = 0 \). Thus, \( R \) is not symmetric. 
3. Transitive: A relation is transitive if \( (x, y) \in R \) and \( (y, z) \in R \) implies \( (x, z) \in R \). We have \( y = 3x \) and \( z = 3y \). Substituting \( y \) from the first equation into the second gives \( z = 3(3x) = 9x \). This means if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) = (x, 9x) \) is also in \( R \). 
Consequently, the relation \( R \) is transitive.