Definition:Cofinal Relation on Ordinals

Definition
Let $x$ and $y$ be ordinals.

Then $x$ and $y$ are said to be cofinal iff there exists a mapping $f: y \to x$ such that:


 * $(1): \quad y \le x$
 * $(2): \quad f$ is strictly increasing.
 * $(3): \quad$ For all $a \in x$, there is some $b \in y$ such that $f \left({b}\right) \ge a$.

Notation
If $x$ and $y$ are cofinal, the notation $\operatorname{cof} \left({ x,y }\right)$ can be used.

Warning
$\operatorname{cof}$ is not commutative. In fact, it is antisymmetric.