Eric Forgy inner product of multisets

See inner product of multisets

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The following discussion began as a query box at multiset

Eric: What is the motivation for the above definition of intersection? I’ve been doodling a lot lately and think a more natural definition for intersection would be

i.e. the intersection of two multisets should just be a set.

When $\mathcal{X}$ and $\mathcal{Y}$ are just sets, we have

A natural generalization of this would seem to be

where the above actually defines the intersection of multisets. However, if we define

then the multiplicity is one and we are back to a simple set. If we were to define intersection this way, then the intersection of two multisets is always a set. This would mean that $\mathcal{X}\cap\mathcal{X}\ne\mathcal{X}$ EXCEPT when $\mathcal{X}$ is a set, in which case we do have $\mathcal{X}\cap\mathcal{X} = \mathcal{X}$.

However, instead of defining intersection in that convoluted way, I would just define

Then the result

would follow. Furthermore, when $\mathcal{X}$ and $\mathcal{Y}$ are just sets, we’d have

which has a certain intuitive feeling to it.

Toby: I have no idea what

even means. In particular, I don't know how to divide (or even multiply) a multiset by a multiplicity function.

Can you work out your proposed definition of $\mathcal{X} \cap \mathcal{Y}$ for the case of $\mathcal{X} = \{1,1,2,3\}$ and $\mathcal{Y} = \{1,1,2,2\}$?

Eric: Ack. What I wrote only has a hope of being “ok” when $\mu_X$ and $\mu_Y$ are constant. When they are constant, I hope the equations make sense.

As far as your example. Good idea! I want to decompose the sum

Thanks. This example illustrates the depths of my confusions. Sorry about that. Your example tells me that we want

In other words,

as expected from the original definition of $\cap$ for multisets.

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Law of Cosines

Warning: This section is somewhat tentative and may need blessing.

In this section, we develop the law of cosines for multisets.

Given the sum of disjoint multisets

note that

However, when $\mathcal{X}$ is merely a set $\langle X,\chi_X\rangle$, then

and we have the familiar expression

Furthermore, when $X$ and $Y$ are sets we can decompose their sum into three disjoint multisets

resulting in

or

which is a law of cosines for sets.

This is different than the traditional way to derive the law of cosines because with multisets we have

and