I had problems with the definition of the empty product. If the upper bound of the product is lower than the lower bound, there is no index in between, so the product is defined as 1. But isn't it possible, like with an integral, to swap the boundaries and make the product negative? Then the product wouldn't be empty?
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1A product is something different than an integral. The definition is made on purpose. We don't want to swap. See this multi-duplicate. This has been asked very often already. – Dietrich Burde Apr 11 '23 at 10:45
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Personally, I've always understood $\prod_{j=a}^bf(j)$ as $\prod\limits_{x\in \Bbb Z\cap [a,b]} f(x)$, and sum likewise. It is quite apparent, at least to me, that there is no argument for doing the manipulation you suggest (whatever that is). – Apr 11 '23 at 10:46
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Is there a difference between an empty product and an product without elements? Might sound silly but I found that they were sometimes defined as different things. – Anouk S Apr 11 '23 at 11:00