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 \( R \) contains pairs of students' roll numbers such that the second roll number is three times the first. 
Thus, if \( x \) is the roll number of a student, then \( y = 3x \) is the roll number of another student. For example, if the roll numbers of the students are \( 1, 2, 3, 4, 5, 6, \dots, 10 \), the elements of \( R \) will be: \[ R = \{ (1, 3), (2, 6), (3, 9) \} \] 
Now, let's check if the relation is reflexive, symmetric, and transitive: 
1. Reflexive: A relation is reflexive if every element is related to itself. For reflexivity, we would need \( (x, x) \in R \) for all \( x \). 
Since \( y = 3x \), it is impossible for \( y = x \), so \( R \) is not reflexive. 
2. Symmetric: A relation is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \in R \). Since \( y = 3x \), there is no corresponding pair \( (y, x) \) where \( x = 3y \), so \( R \) is not symmetric. 
3. Transitive: A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). Since \( y = 3x \) and \( z = 3y = 9x \), we see that \( (x, z) = (x, 9x) \) is also in \( R \). 
Hence, the relation is transitive.