Proving Properties of Relations on Integers

Prove that a given relation $R$ on the set of integers $\mathbb\Z$ is reflexive, symmetric, and transitive.

This problem centers around proving that a relation on the set of integers exhibits three key properties: reflexivity, symmetry, and transitivity. Understanding these properties is fundamental in discrete mathematics, as relations are central to structuring mathematical arguments and defining equivalences.

Reflexivity implies that every element in the set is related to itself. When tackling this, you inspect how the relation applies to each integer, ensuring self-relation. Meanwhile, symmetry requires that if an integer A is related to an integer B, then B must also be related to A. This bidirectional investigation of the relation exposes whether the two-way connectivity holds. Transitivity, on the other hand, is about chaining relations: if an integer A is related to B and B to C, then A should relate directly to C. This demands a close evaluation of the compositional nature of relationships.

Proving these properties not only solidifies comprehension of how relations function but also paves the way for deeper exploration into equivalence relations and partitions of sets. Typically, addressing such problems strengthens skills in logical reasoning and structured proof development, which are invaluable in more advanced mathematical discussions.