I'm new here, so I don't know how to do the fancy symbols. Sorry...
This is for my intro. to adv. math class, and I've been struggling this entire semester. I kinda understand the concept being asked, but I have no idea how to go about proving it.
It was explained that a rational cut is: Let $x$ be an element of $\mathbb{Q}$. The set $\{z \in \mathbb{Q} | z < x\}$ is a rational cut. It is denoted by $x^*$.
My professor defines a Dedekind cut as:
- $\{\alpha,\beta\}$ is a partition of $\mathbb{Q}$, that is,
$$\alpha\cup\beta=\mathbb{Q},\quad \alpha\ne \varnothing\ne\beta,\quad \alpha\cap\beta=\varnothing$$
$\forall a\in\alpha,b\in\beta,\quad a<b$
$\alpha$ has no maximum element in $\mathbb{Q}$. In other words,
$\neg\exists x\in\mathbb{Q}$ such that $\alpha=\{y\in\mathbb{Q}|y\le x\}$
So how can I prove that a rational cut is a Dedekind cut?