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?
Multiset Notation
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multisets
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0$A \in \mathbb{N}^{\{0,1\}}, A(0) + A(1) = \|A\|_1 = n$. $A \in \mathbb{N}^{[1,1000]}$. $A \in \mathbb{N}^\mathbb{R}$. – 2018-06-14
1 Answers
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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.
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0@BrianM.Scott I'm uninformed on this stuff so pardon the simple question, but $S\subseteq \{0,1\}$ means $S=\{1\}$ is possible? But that would rule out any 0's in the multiset? I guess the question(s) is: Can $S$ here be a set with one element. If so, does that just indicate we have a multiset consisting of one element repeated $n$ times? – 2018-04-27