Order relations on sets and fields

#wip Square definitions from Honors Mathematics B and Modern Analysis I.

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Order relations §

Definition: Order relation

An order on a set $X$ is a relation, denoted $<$, which satisfies the following two properties:

• (O1) Trichotomy. Given $x,y\in X$, exactly one of $x<y$ or $x=y$ or $y<x$ is true.
• (O2) Transitivity. If $x<y$ and $y<z$, then $x<z$.

Axioms: Order relations

• (O1) Reflexivity. $x\le x$.
• (O2) Antisymmetry. $x\le y\wedge y\le x⟹x=y$.
• (O3) Transitivity. $x\le y\wedge y\le z⟹x\le z$.
• (O4) Linearity. We must have $x\le y$ or $y\le x$.

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Ordered sets and fields §

Definition: Ordered set

An ordered set $(X,<)$ is a set $X$ together with a relation $<$,

Definition: Ordered field

An ordered field $(\mathbb{F},+,\cdot ,<)$ is a field $(\mathbb{F},+,\cdot )$ with an order $<$ on $\mathbb{F}$ so that we have:

• (OF1) If $x<y$, then $x+z<y+z$.
• (OF2) If $x<y$ and $z>0$ , then $xz<yz$.

• (Lemma) If $\mathbb{F}$ is an ordered field, then the following are true:
  • (a) $x>0⟺-x<0$.
  • (b) $-1<0$ and $1>0$.
  • (c) For any positive natural $n\in \mathbb{N}$ written as a sum $n=1+\cdots +1$, we have $n>0$. In particular, $n\ne 0$, so we can divide by positive integers.
  • (d) Non-negative squares. For any $x\in \mathbb{F}$ we have $x^2\ge 0$, and if $x\ne 0$ then $x^2>0$.

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Review §

• Give an example of an ordered field
• Is ${\mathbb{F}}_p$ orderable? Justify.
• Can the complex numbers $\mathbb{C}$ be made into an ordered field? Justify with a contradiction.
• Justify (OF2) with the property that scaling a positive number by a positive number outputs a positive number. ⭐