Ordered Sets

Let $A$ be a nonempty set. Then $A$ is an ordered set if there is a relation on $A$ denoted by ”<” that satisfies:

1. (Trichotomy) $\forall x,y\in a$ exactly one of the following holds:
   1. $x<y$ or $x>y$ or $x=y$
2. (Transitivity) $\forall x,y,z\in A,x<y\wedge y<z⟹x<z$

Examples:

An ordered set $A$ need not be $A\subset \mathbb{R}$.

• $(\mathbb{Z},<)$
• $(\mathbb{Q},<)$
• $({\mathbb{Z}}^2,<)$ where ”<” is the dictionary ordering.
  • $(m,n)<(m^{\prime },n^{\prime })⟺$ either $m<m^{\prime }\vee (m=m^{\prime }\wedge n<n^{\prime })$. Check first position, then second position.
  • Thus, $(0,1)<(0,2)$.

Notation

$$x\le y$$

means $x<y\vee x=y$.

$$x>y$$

means $y<x$.

Bounded Above

Let $A$ be an ordered set. Let $B\subset A,B=\varnothing$. Then we say $B$ is bounded above if $\exists a\in A$ such that $\forall b\in B,b\le a$.

If $B\subset A,B\ne \varnothing$ is bounded above, then every $a\in A$ satisfies the definition, and is called an upper bound for $B$.

                a------)
<-----(-----(---)------)---->
      A     B   B      A