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June 28th, 2014, 12:30 PM   #1
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Addition with Dedekind cuts, proof

I'm trying to show that $r +_R s$ is a real number, i.e. a Dedekind left set. With this notation $r$ and $s$ are both Dedekind left sets.

I'm using the definition that $r+_Rs$ = {$p+_Qq : p \in r$ and $q \in s$}

In particular, I'm having some difficulty showing $r +_R s$ is closed to the left, i.e. if $q \in {r+_Rs}$ and $p <_Q q$, then $p\in {r+_Rs}$.

What I have done thus far is...

Suppose $a+_Qb \in r+_Rs$. Further suppose that $c,d \in Q$ and $c+_Qd<_Qa+_Qb$.
(*) Then there is some $j \in r$ and $k \in s$ such that $j+_Qk=c+_Qd$.
Since $j \in r$ and $k \in s$, then $j+_Qk$, which is equal to $c+_Qd$, must be in $r+_Rs$. (I feel like I'm stretching the concept of equality a bit here).

Now I made the statement (*), because I thought of a counterexample to the statement: Then $c \in r$ and $d \in s$.

Here's my counterexample: Let $c=-7$ and $d=8$. Then $-7+_Q8=1$ and $1 \in -3+_R4$, but $ 8 \ni 4$ and $8 \ni -3$.

Is the statement (*) legit? It seems very obvious to me, but I'm having a hard time proving it. Please help.

This is more of a set theory question, but since it involved the construction of the reals, I thought it best to place this thread in the real analysis section.
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June 28th, 2014, 08:18 PM   #2
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The problem, as you probably noticed, happens when you add a negative number to a positive one. This makes being precise "conceptually difficult".

So let's just start with:

$p <_{\Bbb Q} q$.

Now $q \in r +_{\Bbb R} s$, so that means:

$q = a +_{\Bbb Q} b$, for $a \in r, b \in s$.

Since $q = p +_{\Bbb Q} (q -_{\Bbb Q} p) = a +_{\Bbb Q} b$

$p = (a + (p -_{\Bbb Q} q)) +_{\Bbb Q} b$.

So we just need to show that $a + (p -_{\Bbb Q} q) \in r$.

But this follows because $a + (p -_{\Bbb Q} q) <_{\Bbb Q} a$

(since $p -_{\Bbb Q} q <_{\Bbb Q} 0$), and $a \in r$.

To make the arguments simpler, it is often easier to define "positive real numbers first", then add the cut 0, then define "formal negatives" (a negative real number is the COMPLEMENT of the closure (in $\Bbb Q$) of the set:

$-r = \{-q: q \in r\}$ where $r$ is a positive real number (that is $r$ contains some positive rational number). We need to take the complement of the closure, instead of just the complement, to handle the case where $r$ contains its least upper bound).
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