I'm reading Pugh's "Real Mathematical Analysis" where C.Pugh constructs real numbers system using Dedekind cuts. Unfortunately, he omits a construction of a multiplicative inverse of a real number $x = A|B$ where $A$ and $B$ are cuts in $\mathbb{Q}$. First, some required definitions:
A cut in $\mathbb{Q}$ is a pair of subsets $A,B$ of $\mathbb{R}$ such that
a)$A \cup B = \mathbb{Q}, A \neq \varnothing, B \neq \varnothing, A \cap B = \varnothing$
b) If $a \in A$ and $b \in B$ then $a < b$
c) A contains no largest element
A real number is a cut in $\mathbb{Q}$.
The additive inverse of a cut $x = A|B$ is defined as $(-x) = C|D$ where $C = \{r \in Q| \exists b \in B$ (not the smallest elements of $B$ ) $: r = -b \}$ and $D = \mathbb{Q} \setminus C$
Multiplication is defined for positive numbers $x > 0$. If $x = A|B, y = C|D$, then $xy = E|F$ where
$E = \{r \in \mathbb{Q}| r \leq 0$ or $\exists a \in A \ \ \exists c \in C: a > 0, c > 0, r = ac \}, F = \mathbb{Q} \setminus E$. If $x > 0, y < 0$ then we define $xy = -(x(-y))$
Now I wonder how can we define a multiplicative inverse of a positive cut $x = A|B$. Obviously, it's a cut $\frac{1}{x} = C|D$ such that $x \frac{1}{x} = 1$
So, it's such a cut $C|D$ such that for $C$ we have
$\{r \in \mathbb{Q}| r \leq 0$ or $\exists a \in A \ \ \exists c \in C : a > 0, b > 0, r = ac \} = \{r \in \mathbb{Q}| r < 1 \}$
Any ideas how we can define $C$(if we define $C$, then $D = \mathbb{Q} \setminus C$)?
Best Answer
Edit: Changed to correct the error modnar points out in the comments.
If $x > 0$, $$C = \{r \in \Bbb Q \mid \exists b \in B,\, rb < 1\}$$ If $x < 0$, $$C = \{r \in \Bbb Q \mid \exists b \in B,\, b < 0\text{ and } rb > 1\}$$
Original post: (has maximums when $x$ is rational)
If $x > 0$, $$C = \{ r \in \Bbb Q \mid r \le 0 \text{ or } \forall a \in A,\, ra < 1\}$$
If $x < 0$, $$C = \{ r \in \Bbb Q \mid \forall a \in A, \,ra > 1\}$$