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

Notation
The statement "$x$ and $y$ are cofinal" is sometimes written $\operatorname{cof} \left({ x,y }\right)$.