Which concept is more general, Cartesian Product or Rule of Product or Direct Product? This article about the Rule of Product and this article about Cartesian Product seem to be about the same thing -- at least the n-arity part of the Cartesian product. Then there is the Direct Product. All of these seem to be saying similar things. Which is the most fundamental? Which is a derivation of which? Or are they just different labels on the same idea?
 A: I cannot act as an authority in terminology, but I've had a look at what the Wikipedia says, and the idea is this:

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*For sets, you have Cartesian product, which may be finite (a finite number of sets, the members are tuples) but may be infinite as well (the members are functions).

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*For the Rule of Product, only finite Cartesian products are relevant, as it is all about counting the sizes of sets, and multiplying them out.



*Then, direct product is an extension of Cartesian product for sets endowed with additional structure, e.g. operations, relations, subsets etc. It coincides with the Cartesian product if your sets are just "plain" sets, without any additional structure. In the other cases it normally has the Cartesian product of the sets as the underlying set of the direct product. The onus here is on defining the additional structure on that.

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*However, you have to define the direct product differently depending on the additional structure. For instance, the direct product of groups and of topological spaces have each their own definitions and it is not immediately clear how (and whether at all!) they are related.



*Finally, the category theory provides a unique framework which explains all those direct products (and the reasons why they are defined as they are) and also lets you define direct products for any new category of things that you may come up with!

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*In the category of sets, the category theory product coincides with both direct product and Cartesian product.



