Question

Show that the relation:

\[ R = \{(a, b) : (a - b) \text{ is a multiple of 5} \} \]

on the set \( \mathbb{Z} \) of integers is an equivalence relation.

Hint:

For equivalence relations, always verify reflexivity, symmetry, and transitivity step-by-step.

Answer 1

To prove \( R \) is an equivalence relation, we verify the three properties:

1. Reflexive: For any \( a \in \mathbb{Z} \), \( a - a = 0 \), which is a multiple of 5. Hence, \( (a, a) \in R \).
2. Symmetric: If \( (a, b) \in R \), then \( a - b \) is a multiple of 5. Thus, \( b - a = -(a - b) \), which is also a multiple of 5. Hence, \( (b, a) \in R \).
3. Transitive: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( a - b \) and \( b - c \) are multiples of 5. Adding these gives \( a - c = (a - b) + (b - c) \), which is a multiple of 5. Hence, \( (a, c) \in R \).

Since \( R \) satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.