multiset
Note that the standard definition of a set also allows repeated elements, but these are not treated as repeated elements. For example, $\{1,1,3\}$ as a set is actually equal to $\{1,3\}$. However, as a multiset, $\{1,1,3\}$ is not simplifiable further.
A definition that makes clear the distinction between set and multiset follows:
Multiset.
A multiset is a pair $(X,f)$, where $X$ is a set, and $f$ is a function mapping $X$ to the cardinal numbers^{} greater than zero. $X$ is called the underlying set of the multiset, and for any $x\in X$, $f(x)$ is the multiplicity of $x$.
Using this definition and expressing $f$ as a set of ordered pairs^{}, we see that the multiset $\{1,3\}$ has $X=\{1,3\}$ and $f=\{(1,1),(3,1)\}$. By contrast, the multiset $\{1,1,3\}$ has $X=\{1,3\}$ and $f=\{(1,2),(3,1)\}$.
Generally, a multiplicity of zero is not allowed, but a few mathematicians do allow for it, such as Bogart and Stanley. It is far more common to disallow infinite^{} multiplicity, which greatly complicates the definition of operations^{} such as unions, intersections^{}, complements, etc.
References
- 1 Kenneth P. Bogart, Introductory Combinatorics. Florence, Kentucky: Cengage Learning (2000): 93
- 2 John L. Hickman, “A note on the concept^{} of multiset” Bulletin of the Australian Mathematical Society 22 (1980): 211 - 217
- 3 Richard P. Stanley, Enumerative Combinatorics Vol 1. Cambridge: Cambridge University Press (1997): 15
Title | multiset |
---|---|
Canonical name | Multiset |
Date of creation | 2013-03-22 12:21:44 |
Last modified on | 2013-03-22 12:21:44 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 13 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 03E99 |
Synonym | bag |
Related topic | AxiomsOfMetacategoriesAndSupercategories |
Related topic | ETAS |
Defines | multiplicity |