Multiset Notation

Question

There is a multiset $A$, of length $n$ that can contain only $1s$ or $0s$. How would I notate that? How about for a multiset that could contain any number from $1-1000$, or that could contain any real number?

Answer 1

A multiset of length $n$ that contains only $0$s and $1$s can be described as a multiset of cardinality $n$ whose underlying set is a subset of $\{0,1\}$. If you need to be formal, this is an ordered pair $\langle S,m \rangle$ such that $S \subseteq \{0,1\}$, $m:S \to \mathbb{Z}^+$, and $\sum\limits_{s \in S}m(s) = n$. Here $S$ is the underlying set, and $m$ is the multiplicity function. You can of course replace $\{0,1\}$ by any other set: $\{n \in \mathbb{Z}:1 \le n \le 1000\}$, $\mathbb{R}$, or whatever.