Prove $\sim$ is an equivalence relation: $x \sim y$ if and only if $y = 3^kx$, where $k$ is a real number The relation $\sim$ is defined on $\mathbb{Z}^+$ (all positive integers). We say $x\sim y$ if and only if $y=3^kx$ for some real number $k$.
I need to prove that $\sim$ is an equivalence relation.
To prove an equivalence relation we must certify that:


*

*It is reflexive

*It is symmetric

*It is transitive


I am not sure how to start this off since if I want to prove for:
Reflexive: I would replace $y$ with $x$, so that $x = (3^k)*x$ which is a positive integer when $k > 0$. Thus $x \sim x$ and $\sim$ is reflexive (?).
Symmetric: Suppose $x \sim y$ so that $y = (3^k)*x$ then $x = (3^k)*y$ is also an element of $\mathbb{Z}^+$. Thus $y \sim x$ and $\sim$ is symmetric (?).
Transitive: I'm not too sure about this last one. I think my entire proof is wrong anyways.
Can anyone help me on this?

Isn't this relation true for all $x,y$ with the same sign? Since $3^k$, $k$ in reals, can take any positive real number?
