Understanding products and powers

[RaphaelS1]

Copied from email to Franz

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Quick follow up question about product sets, it seems to me that if you take the product of a product then you result in a set containing tuples of tuples and sets, so by example:

{1, 2} * {1,2} = {(1,1), (1,2), (2,1), (2,2)}
{(1,1), (1,2), (2,1), (2,2)} * {1,2} = {((1,1),1), ((1,1),2), ((1,2),1), ((1,2),2),….,}

However this seems counterintuitive to me as this then means that (1, 1, 1) is not in {1, 2}^3 but ((1, 1), 1) is, which are not the same!

Have I missed a property of sets/tuples? Or just overlooking something in the Cartesian product?

[fkiraly]

also copying reply, so it's in one place.

"What you say in the pre-amble, i.e.,
{1, 2} * {1,2} = {(1,1), (1,2), (2,1), (2,2)}
{(1,1), (1,2), (2,1), (2,2)} * {1,2} = {((1,1),1), ((1,1),2), ((1,2),1), ((1,2),2),….,}

is entirely correct.

Formally, there is a difference between the Cartesian product between three sets, i.e.,
\bigtimes_{n=1}^3 \NN = \NN^3
which contains elements such as (1,2,3) and (3,5,42), and iterating this twice, i.e.,
(\NN \times \NN) \ times \NN,
which contains elements such as ((1,2),3) and ((3,5),42).

The reason why this is usually brushed over is that there is a so-called “canonical isomorphism” between the two, i.e., a bijection which maps (1,2,3) to ((1,2),3) and vice versa, and (3,5,42) to ((3,5),42) and vice versa.

However, you are correct in spotting: unlike in schoolbook multiplication (which is associative and commutative, and where only the result matters), the bracketing matters in symbolic expressions. Furthermore, the bracketing in the Cartesian product is replicated by the bracketing of tuples.

For the interface, I think it means we should make the distinction, too – by allowing the “shallow” product of n sets, which replicates the behaviour of the \bigtimes operator (no inner bracketing), and by allowing to nest them.

[fkiraly]

PS: I don't think any of the content is a "power", as the title alludes?

[RaphaelS1]

Well {1, 2} * {1,2} == {1.2}^2

[fkiraly]

ah, those powers, i.e., "n-ary Cartesian power".
Funny clash of terminology, compare "power set" which is not closely related.

[RaphaelS1]

• Add nest argument to product
• Default is !nest
• If nest then don't unwrap wrappedSets
• If !nest then unwrap all the sets and call product on a list of (c(x,y))