multiset

A multiset is a set for which repeated elements are considered.

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.