Two increasing functions from the set of positive integers to the set of positive integers such that neither f (n) is O(g(n)) nor g(n) is O(f (n))
Take $f(n) = (2\lfloor\frac{n}{2}\rfloor+1)!$ and $g(n) = (2\lceil \frac{n}{2} \rceil)!$.
For example:
$$
\begin{array}{c|ccccc}
n&amp;1&amp;2&amp;3&amp;4&amp;5 \\\hline
f(n)&amp; 1! &amp; 3! &amp; 3! &amp; 5! &amp; 5! \\\hline
g(n)&amp; 2! &amp; 2! &amp; 4! &amp; 4! &amp; 6!
\end{array}
$$
We have
$$
\frac{f(2m)}{g(2m)} = \frac{(2m+1)!}{(2m)!} = 2m+1 \to \infty
$$
as well as
$$
\frac{f(2m+1)}{g(2m+1)} = \frac{(2m+1)!}{(2m+2)!} = \frac{1}{2m+2} \to 0.
$$