Equivalence Relation on Natural Numbers such that Quotient is Power of Two

Theorem
Let $\alpha$ denote the relation defined on the natural numbers $\N$ by:
 * $\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$

Then $\alpha$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
We have that for all $x \in \N$:


 * $x = 2^0 x$

and as $0 \in \Z$ it follows that:
 * $x \mathrel \alpha x$

Thus $\alpha$ is seen to be reflexive.

Symmetry
Thus $\alpha$ is seen to be symmetric.

Transitivity
Let $x \mathrel \alpha y$ and $y \mathrel \alpha z$.

Then by definition:

Thus $\alpha$ is seen to be transitive.

$\alpha$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.