Set of Cuts forms Ordered Field
Theorem
Let $\CC$ denote the set of cuts.
Let $\struct {\CC, +, \times, \le}$ denote the ordered structure formed from $\CC$ and:
- the operation $+$ of addition of cuts
- the operation $\times$ of multiplication of cuts
- the ordering $\le$ of cuts.
Then $\struct {\CC, +, \times, \le}$ is an ordered field.
Proof
First we show that $\struct {\CC, +, \times}$ is a field by demonstrating that it fulfils the field axioms:
\((\text A 0)\) | $:$ | Closure under addition | \(\ds \forall x, y \in F:\) | \(\ds x + y \in F \) | |||||
\((\text A 1)\) | $:$ | Associativity of addition | \(\ds \forall x, y, z \in F:\) | \(\ds \paren {x + y} + z = x + \paren {y + z} \) | |||||
\((\text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall x, y \in F:\) | \(\ds x + y = y + x \) | |||||
\((\text A 3)\) | $:$ | Identity element for addition | \(\ds \exists 0_F \in F: \forall x \in F:\) | \(\ds x + 0_F = x = 0_F + x \) | $0_F$ is called the zero | ||||
\((\text A 4)\) | $:$ | Inverse elements for addition | \(\ds \forall x \in F: \exists x' \in F:\) | \(\ds x + x' = 0_F = x' + x \) | $x'$ is called a negative element | ||||
\((\text M 0)\) | $:$ | Closure under product | \(\ds \forall x, y \in F:\) | \(\ds x \times y \in F \) | |||||
\((\text M 1)\) | $:$ | Associativity of product | \(\ds \forall x, y, z \in F:\) | \(\ds \paren {x \times y} \times z = x \times \paren {y \times z} \) | |||||
\((\text M 2)\) | $:$ | Commutativity of product | \(\ds \forall x, y \in F:\) | \(\ds x \times y = y \times x \) | |||||
\((\text M 3)\) | $:$ | Identity element for product | \(\ds \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:\) | \(\ds x \times 1_F = x = 1_F \times x \) | $1_F$ is called the unity | ||||
\((\text M 4)\) | $:$ | Inverse elements for product | \(\ds \forall x \in F^*: \exists x^{-1} \in F^*:\) | \(\ds x \times x^{-1} = 1_F = x^{-1} \times x \) | |||||
\((\text D)\) | $:$ | Product is distributive over addition | \(\ds \forall x, y, z \in F:\) | \(\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} \) |
These are called the field axioms.
It has been established from Set of Cuts under Addition forms Abelian Group that $\struct {\CC, +}$ forms an abelian group.
Thus $\text A 0$ through to $\text A 4$ are fulfilled.
It remains to verify that $\struct {\CC, +, \times}$ fulfils the remaining field axioms.
Field Axiom $\text M0$: Closure under Product
From Product of Cuts is Cut:
- $\forall \alpha, \beta \in \CC: \alpha \beta \in \CC$
Thus $\struct {\CC, \times}$ is closed.
$\Box$
Field Axiom $\text M1$: Associativity of Product
From Multiplication of Cuts is Associative:
- $\paren {\alpha \beta} \gamma = \alpha \paren {\beta \gamma}$
Thus the operation $\times$ of multiplication of cuts is associative on $\struct {\CC, \times}$.
$\Box$
Field Axiom $\text M2$: Commutativity of Product
From Multiplication of Cuts is Commutative:
- $\alpha \beta = \beta \alpha$
Thus the operation $\times$ of multiplication of cuts is commutative on $\struct {\CC, \times}$.
$\Box$
Field Axiom $\text M3$: Identity for Product
Consider the rational cut $1^*$ associated with the (rational) number $1$:
- $1^* = \set {r \in \Q: r < 1}$
From Cut Associated with 1 is Identity for Multiplication of Cuts:
- $\alpha \times 1^* = \alpha$
for all $\alpha \in \CC$.
From Multiplication of Cuts is Commutative it follows that:
- $1^* \times \alpha = \alpha$
That is, $1^*$ is the identity element of $\struct {\CC, \times}$.
$\Box$
Field Axiom $\text M4$: Inverses for Product
We have that $1^*$ is the identity element of $\struct {\CC, \times}$.
Let $\alpha \ne 0^*$.
From Existence of Unique Inverse Element for Multiplication of Cuts, there exists a unique cut $\dfrac 1 \alpha$ such that:
- $\alpha \times \dfrac 1 \alpha = 1^*$
From Multiplication of Cuts is Commutative it follows that:
- $\dfrac 1 \alpha \times \alpha = 1^*$
Thus every element $\alpha$ of $\struct {\CC, \times}$ such that $\alpha \ne 0^*$ has an inverse $\dfrac 1 \alpha$.
$\Box$
Field Axiom $\text D$: Distributivity of Product over Addition
From Multiplication of Cuts Distributes over Addition:
- $\alpha \paren {\beta + \gamma} = \alpha \beta + \alpha \gamma$
for all $\alpha$, $\beta$ and $\gamma$ in $\CC$.
$\Box$
All the field axioms are thus seen to be fulfilled, and so $\struct {\CC, +, \times}$ is a field.
$\Box$
Finally it is shown that $<$ is a total ordering on $\struct {\CC, +, \times}$:
This is demonstrated in Ordering on Cuts is Total.
From Properties of Ordered Field, this is all that is required to be shown.
$\blacksquare$
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers