I wanted to receive some help regarding some homework questions; I appreciate any sort of feedback possible.
I am certain that any Dedekind Cut $A$ has the following properties:
1) $A$ is not the empty set and is NOT the set $\mathbb Q$. 2) For any $a \in A$ and $b$ $\in \mathbb Q$ \ A, $a
Working on defining addition and multiplication for Dedekind Cuts (as the set of all dedekind cuts with the order $\subseteq$ is the set $\mathbb R$.
The additive identity is defined as: $0_\mathbb R$ $:=$ {$y$ $\in \mathbb Q$ such that $y < 0$}.
Question 1: I need to prove that for any Dedekind Cuts $X$ and $Y$, that $X$ + $0_\mathbb R$ $= X$.
This requires me to show that they are both subsets of each other (to be true). By the definition of addition of Dedekind cuts, we have:
$X$ + $0_\mathbb R$ $:=$ {$z$: $z \in \mathbb Q$ such that $z=x+y$, where $x \in X$ and $y \in$ $0_\mathbb R$}.