Relations - Very basic notation

Question

I'm struggling to understand some basic notation regarding relations. Say I'm given some relation Z × (Z \ {0}) "on" (a,b)R(x,y) if and only if ab=xy, what exactly is this asking? and is this an equivalence relation?

Answer 1

I think you might mean that $R$ is a relation on $\mathbb{Z} \times \mathbb{Z} \setminus \{0\}$. Assuming this, a relation $R$ on $\mathbb{Z} \times \mathbb{Z} \setminus \{0\}$ is just a subset of $(\mathbb{Z} \times \mathbb{Z} \setminus \{0\}) \times (\mathbb{Z} \times \mathbb{Z} \setminus \{0\})$. When we say that two elements of your set, say $(x,y)$ and $(w,z)$, are related, we sometimes write $(x,y)R(w,z)$.

Now to be an equivalence relation, your relation must be:

1) Reflexive: For all $(x,y) \in \mathbb{Z} \times \mathbb{Z} \setminus \{0\}$, $$(x,y)R(x,y).$$

2) Symmetric: For all $(x,y),(w,z) \in \mathbb{Z} \times \mathbb{Z} \setminus \{0\}$, $$(x,y)R(w,z) \Rightarrow (w,z)R(x,y).$$

3) Transitive: For all $(x,y),(w,z),(u,v) \in \mathbb{Z} \times \mathbb{Z} \setminus \{0\}$, $$(x,y)R(w,z), \text{ and } (w,z)R(u,v) \Rightarrow (x,y)R(u,v).$$

Answer 2

This means that $R$ is a relation on $\mathbb{Z} \times \mathbb{Z}$-{0} which maps an ordered pair (a,b) $\in$ $\mathbb{Z} \times \mathbb{Z}$-{0} to an ordered pair (x,y) $\in$ $\mathbb{Z} \times \mathbb{Z}$-{0} iff the condition $ab=xy$ is satisfied.

And this relation is not equivalent since it is reflexive and symmetric but not transitive. Try to prove it yourself following the rules, if needed, stated by Ebearr.