# Definition:Cofinal Relation on Ordinals

## Definition

Let $x$ and $y$ be ordinals.

Then $y$ is said to be cofinal with respect to $x$ if and only if 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 $y$ is cofinal with $x$, the notation $\operatorname{cof} \left({x, y}\right)$ can be used.

## Warning

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