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?
Best Answer
You are misunderstanding what needs to be shown in order to show that something is an equivalence relation. None of the arguments you present is a valid argument for the proposition you are trying to establish.
Your equivalence relation is defined on the set of positive integers. So if you write $x\sim y$, you are already assuming that $x$ and $y$ are positive integers, there is no need to prove that they are.
The relation is defined as follows: if $x$ and $y$ are positive integers, then $$x\sim y\Longleftrightarrow \text{there exists a real number }k\text{ such that }y = 3^kx.$$
That means that in order to show that two given numbers $x$ and $y$ are related, then you need to produce a real number $k$ that witnesses the identity $y=3^kx$. For example, to show that $y=9$ and $x=3$ are related, I just need to say: "take $k=1$. Then $9 = 3^1\cdot 3$; that is, $y = 3^1x$, so $x\sim y$ holds." To show that $18\sim 6$, I say "take $k=-1$; then $6 = 3^{-1}(18)$ holds." Etc.
And if you know that $x\sim y$, then you know that there exists a real number $k$ such that $y=3^kx$.
Thus, to show that $\sim$ is reflexive, you need to show that given any positive integer $x$, you can find a real number $k$ (which may depend on $x$) such that $x = 3^k x$. You need to say who $k$ is. Your argument about "being positive when $k\gt 0$" doesn't get you there in any way.
To show that $\sim$ is symmetric, you have to show that if you already know that $x\sim y$, so that you know there is a real number $k$ such that $y=3^k x$, then you can find some real number $\ell$ such that $x=3^{\ell}y$. This will witness the fact that $y\sim x$ holds, showing symmetry. You already know that $x$ and $y$ are positive integers. You know that simply because you know that $x\sim y$ holds, and that means that $x$ and $y$ have to be positive integers in the first place.
To show that $\sim$ is transitive, you have to show that if you already know $x\sim y$ and $y\sim z$, then you can exhibit a real number $r$ such that $z=3^rx$; this will witness $x\sim z$. You know there is a real number $k$ such that $y=3^kx$ (because $x\sim y$); and you know there is a real number $\ell$ such that $z=3^{\ell}y$ (because $y\sim z$); now you need to produce that number $r$ somehow. Again, you already know that $x$, $y$, and $z$ are positive integers, because you already know that $x\sim y$ and $y\sim z$ are true, which means, inter alia, that $x$, $y$, and $z$ are all positive integers.