# Functional vs Arithmetic Parity

I was wondering today about the relation between evens and odds in regular arithmetic compared with that of even and odd functions. We all know the multiplication table for arithmetic multiplication below (where $e$ is even, $o$ is odd).

\begin{array}{c|cc} \cdot &e &o \\ \hline e& e & e \\ o& e& o \\ \end{array}

What seems strange to me is that if we compare this with the multiplication of even and odd functions we have that if $f$ is an even function so that $f(-x) = f(x)$, and if $g$ is an odd function such that $g(-x) = -g(x)$. So the multiplication seems to be

\begin{array}{c|cc} \cdot &f &g \\ \hline f& f & g \\ g& g& f \\ \end{array}

I dont really see much relation between the multiplication tables despite the naming conventions chosen.

Are the two ideas not really related? Is there some more global definition of parity that I am missing? Can this example be driven further?

Just looking for some basic information about why the properties of even and odd seem to differ so greatly in these two contexts.