Proving Equivalence Relation on Integers

Let $R$ be a relation on the set of integers defined by $a \, R \, b$ if and only if $a - b$ is an integer. Prove that $R$ is an equivalence relation.

This problem involves demonstrating that a given relation, defined specifically on the integers, is an equivalence relation. An equivalence relation has three core properties: reflexivity, symmetry, and transitivity. Recognizing these properties requires understanding how they apply to the set of integers under the defined operation of subtraction. Reflexivity ensures every element relates to itself, symmetry involves the relation being reversible between any two elements, and transitivity means that if the relation holds for pairs of elements in succession, it holds across the span of these elements.

Solving this problem provides insight into the structure and classification of relations, an important concept in discrete mathematics and logic. Equivalence relations often partition sets into equivalence classes, an important outcome when considering problems in symmetry, classification, and modular arithmetic. Identifying an equivalence relation is a primary step in understanding how elements relate to each other under various mathematical frameworks, making it essential knowledge for diving deeper into number theory and more abstract algebraic structures.