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 and 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 EXCEPT when is a set, in which case we do have .
However, instead of defining intersection in that convoluted way, I would just define
Then the result
would follow. Furthermore, when and 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 for the case of and ?
Eric: Ack. What I wrote only has a hope of being “ok” when and 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 for multisets.
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 is merely a set , then
and we have the familiar expression
Furthermore, when and 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
Revised on October 28, 2009 at 18:43:43
by
Toby Bartels