HW2 Problem 14

Determine whether the given relation is an equivalence relation. If it is, describe the partition arising from the equivalence relation.

(a) $n\mathcal{R} m$ in $\mathbb{Z}$ if $nm > 0$

Solution:

Reflexive:
Let $n\in\mathbb{Z}$. Now, $n \cdot n > 0$ is false, because $0 \cdot 0 \ngtr 0$, and $0 \in \mathbb{Z}$. Therefore, the relation is not reflexive.

Hence, $n\mathcal{R} m$ is not an equivalence relation.