# 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 $f=\{(1,2),(3,1)\}$.

## References

Title multiset Multiset 2013-03-22 12:21:44 2013-03-22 12:21:44 PrimeFan (13766) PrimeFan (13766) 13 PrimeFan (13766) Definition msc 03E99 bag AxiomsOfMetacategoriesAndSupercategories ETAS multiplicity