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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.

Thanks for your thoughts.

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4  
The multiplication table for even and odd functions matches the addition table for even and odd numbers. The multiplication table for even and odd numbers matches the true/false table of the OR function. –  Ross Millikan Nov 26 '12 at 21:09
    
The product of two odd functions is even! –  Hagen von Eitzen Nov 26 '12 at 21:38
2  
The composition table for even/odd functions matches the multiplication table for even/odd numbers. –  Joel Cohen Nov 26 '12 at 21:42

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