Question

A relation \( R \) on set \( A = \{1, 2, 3, 4, 5\} \) is defined as \[ R = \{(x, y) : |x^2 - y^2| <8\}. \] Check whether the relation \( R \) is reflexive, symmetric, and transitive.

Hint:

To check reflexivity, symmetry, and transitivity, verify the conditions for all elements in the relation.

Answer 1

Reflexive: A relation is reflexive if for every element \( x \in A \), \( (x, x) \in R \). Since \( |x^2 - x^2| = 0 \), which is less than 8, we conclude that \( (x, x) \in R \) for all \( x \in A \). Thus, the relation is reflexive. Symmetric: A relation is symmetric if for every pair \( (x, y) \in R \), the pair \( (y, x) \) is also in \( R \). Since \( |x^2 - y^2| <8 \) implies \( |y^2 - x^2| <8 \), the relation is symmetric. Transitive: A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). Consider the elements \( x = 1, y = 2, z = 3 \): \[ |1^2 - 2^2| = |1 - 4| = 3 <8, \quad |2^2 - 3^2| = |4 - 9| = 5 <8. \] However: \[ |1^2 - 3^2| = |1 - 9| = 8 \not < 8. \] Thus, the relation is not transitive. Final Answer: The relation \( R \) is reflexive and symmetric, but not transitive.