Determining if $R=\left\{(f,g)\mid \exists k\in\Bbb Z,\forall x\in\Bbb Z, \ f(x)g(x)\lt k\right\}$ is an equivalence

Question

I need help with proving whether or not the relation $R=\left\{(f,g)\mid \exists k\in\Bbb Z,\forall x\in\Bbb Z, \ f(x)g(x)\lt k\right\}$ is an equivalence relation on $\mathbb{Z}\times\mathbb{Z}$

I understand the basic key components needed in order to determine if the relation is an equivalence relation, like reflexivity, symmetry, and transitivity but i can't seem to find a way to proof reflexivity.

Answer 1

Consider the set $A$ of all functions $f:\mathbb{Z}\to\mathbb{Z}$. Define an equivalence relation on $A$ by:

$$R=\{(f,g)|\exists k,\forall x,f(x)g(x)<k\}\subset A\times A$$

I claim the relation is symmetric, but not reflexive or transitive. For symmetry, it suffices to note that $f(x)g(x)=g(x)f(x)$.

As a counterexample to reflexivity, consider $f(x)=x$, and note that $(f,f)\notin R$ because $x^2$ is not bounded.

Finally, consider the functions $f(x)=h(x)=x$ and $g(x)=0$. Then we have $(f,g),(g,h)\in R$, but $(f,h)\notin R$ so the relation is not transitive.

It follows that the relation is not an equivalence relation.