Exercise A0.V.1.11

Proposition

Let $a \sim b$ stand for ”$a$ and $b$ are associates,” that is, $(a) = (b).$ Then $\sim$ is an equivalence relation.

Proof

Certainly $(a) = (a)$ so $\sim$ is reflexive. By symmetry of equality $(a) = (b)$ implies $(b) = (a),$ so $\sim$ is symmetric. Suppose $(a) = (b)$ and $(b) = (c).$ Then $(a) = (c)$ by transitivity of equality. So $\sim$ is a reflexive, symmetric, transitive relation. Hence, it is an equivalence relation.

$\blacksquare$