Total Order Relations

Definition

If $R$ is a partial order relation on set $A$, and for any two elements $x$ and $y$ in $A$, either $xRy$ or $yRx$, then $R$ is a total order relation (or total order) on $A$ $R$ is a total order iff $R$ is a partial order and $\forall x,y\in A(xRy\vee yRx)$ Also called a linear order

Example

The $\le$ relation on $\mathbb{Q}$ is a total order because $\forall x,y\in \mathbb{Q}(x\le y\vee y\le x)$

Well-Ordered Set

Let $\preccurlyeq$ be a total order on a set $A$. $A$ is well-ordered iff every non-empty subset of $A$ contains a smallest element $\forall S\in \mathcal{P}(A),S\ne \varnothing \Rightarrow (\exists x\in S\forall y\in S(x\preccurlyeq y))$