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multiset

Defines: 
multiplicity
Synonym: 
bag
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

03E99 no label found

Comments

Hi, I've some suggestions to improve the multiset definiton.

1) Multiset is also known as bag
2) in the formal definition represent f:X\to{1,2,3,...} - avoid the use of "function mapping"
3) in the formal definition say, "A multiset M is a pair..." and conclude saying "f(x) if the multiplicity of x in M"
4) stress two possible way of representing a multiset, as sequence of (element,multiplicity) pairs or as collection of objects, each repeated with its multiplicity {...}

An observaton on suggestion 2:

The problem with changing the formal definition would be that f:X\to{1,2,3,...} is not the same thing as f being a cardinal-valued mapping. A mapping to the cardinal numbers (or even to the nonzero cardinals) is more general in that it allows for "infinite" multiplicities (\aleph_0, \aleph_1, etc.) that are not permitted in the mapping to the positive integers.

A question regarding suggestion 4:

I'm a bit confused by this, since I don't think every multiset can actually be represented as a *sequence* of (element, multiplicity) pairs. For example, if M = (R+, f) where R+ are the positive real numbers and f(x) = ceil(x), this should be a valid multiset, but cannot be represented as a sequence. As far as I am aware, either of these representations only works for finite multisets. Am I missing something?

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