Determine whether the given relation is an equivalence relation on the set.
$n$ is related to $m$ in the set of integers if $nm>0$.
So my teacher said this set is not an equivalence relation because it is not reflexive. It is not reflexive because $0$ is not related to $0$. Can someone explain how $0$ is not related to $0$?
Best Answer
The relation is $R = \{(n,m) \; | \; nm > 0\}$ where $n,m \in \mathbb{Z}$. But $(0,0) \notin R$ because $0 \cdot 0 \not> 0$, so $0$ is not related to $0$. In order for a relation $R$ on a set $A$ to be reflexive, we must have that $(x,x) \in R$ for all $x \in A$. But for this example, we have a relation on $\mathbb{Z}$, where $0 \in \mathbb{Z}$, but $(0,0) \notin R$, so it is not reflexive.