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How is even and odd related to + and - because 2 odd functions or 2 even functions (and permutation groups too) are multiplied together to give an even result. If they are different like odd even then we get odd exactly like how scalars with the same sign gives a positive and numbers with opposite signs gives a negative. If even is analogous to positive and odd is analogous to negative then what would be analogous to neither even nor odd (0)… In fact can the concept of even and odd somewhat in a clever way be extended to reals such as 1.253?

A lot of cool thoughts to explore possible extensions. Would like some help with this because I am (relatively speaking) a newbie compared to graduates.

Captain HD
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  • You’re title made me think you were talking about even and odd numbers. – Thomas Andrews Sep 29 '21 at 17:15
  • In this post I am trying to link together many different things such as the idea of extending even and odd to reals but also noticing the analogy between positives and negatives to the idea of even and odd such as with functions or in group theory – Captain HD Sep 29 '21 at 17:17
  • Please don't edit back in those tags I deleted. This question has nothing to do with [tag:group-theory] for instance. – Shaun Sep 29 '21 at 17:21
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    This question certainly has to do with group theory. What Captain HD is asking boils down to asking what sets $S$ with multiplication are homomorphic to $\mathbb{Z}/2\mathbb{Z}$? – While I Am Sep 29 '21 at 17:26
  • See my answer to the question https://math.stackexchange.com/questions/3945897/why-it-is-important-to-write-a-function-as-sum-of-even-and-odd-functions/3945955. – KCd Sep 30 '21 at 19:30

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Functions behave differently under multiplication and composition.

When you multiply two even functions the result is an even function

When you multiply two odd functions the result is an even function

When you multiply an odd and an even function the result is an odd function.

When you compose two even functions the result is an even function.

When you compose two odd functions the result is an odd function.

When you compose an odd and an even function the result is an even function.

You can find similarities between functions behavior with addition and muliplication of integers

Mohammad Riazi-Kermani
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  • I know this already. I said the same (without the mention of composition) already in the post but I am looking for connections and deep insights – Captain HD Sep 30 '21 at 19:27