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Questions tagged [multisets]

For questions about or related to multisets, a notion similar to sets with the difference that elements can be repeated.

A multiset, also called a bag, is a generalization of a set in which elements can appear more than once. Formally, a multiset can be defined as a pair $(A, m)$ where $A$ is a set and $$m : A \to \{1, 2, 3, ...\}$$

For each $a \in A$, the number $m(a)$ is called the multiplicity of $a$, and counts how many times $a$ appears in the multiset.

Reference: Multiset.

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Is there such a thing as a multiset with a "negative" number of some element?

Is it possible for a multiset to have a "negative" number of one or more elements? If so, how are such multisets defined, and what terminology exists for them?
J.P.
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Combinations of multisets - the theory?

I've read over the theory countless times, and I still have no idea how to think of it. The formula for the combinations of multisets is $C(k + r - 1, r)$, where $k$ = the number of distinct elements, and $r$ is the $r$-combinations required.…
user1746848
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How to define a map from a multiset to a multiset

Multisets are like sets, but can contain duplicates. Suppose we have two multisets $A, B$, and we want to define a map from one to the other ie: $$ f : A \rightarrow B.$$ How do we do this? Is it possible to assign one "element" of the multiset…
Ben Sprott
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Why are there two different definitions of multiset union

I just realized that you can use two different definitions of multiset union: First definition: (M⊎N)(x)=M(x)+N(x) Second definition: (M⋓N)(x)=max(M(x),N(x)) I understand the difference between them but I'm not sure I understand why these two…
jep.d
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Verify one of DeMorgan’s Laws for sets

Verify one of DeMorgan’s Laws for sets: $$\bigcap \{S\setminus U:U \in \mathcal U\} = S \setminus \bigcup \{U :U \in \mathcal U\}.$$ Can anyonw show me how to do this? a little confused, thanks
Pasie15
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Multiset: notation for size and number of unique items

Given a multiset, e. g. S = {1, 1, 2, 3, 4, 4, 5}, what would be a short, concise notation to express the number of unique items in the multiset? (five in the given example).
qqilihq
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Distributive property for signed multiset

I'm reading through Wayne Blizard's introduction to signed multisets, which says at the bottom of page 9 that additive union distributes over intersection. I see how this is true for normal (non-negative multiplicity) multisets, but for signed…
Johannes Riecken
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Notation for sum over a multiset

Let $\mathcal M$ be a multiset and $\odot$ an associative operation over the support set $M$ of $\mathcal M$. Further, denote by $\mu_{\mathcal M}(x)$ the multiplicity of the element $x$ in $\mathcal M$. As usual in basic multiset theory, the…
user2888243
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Solving problem involving intersection of sets

In a certain office,one-quarter of the staff are left-handed. One-twelfth of them are left-handed and short-sighted; 13 are short sighted while 17 are neither left handed nor short-sighted.Find the number of staff in the office.
San San
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What is a partial multiset?

A. Rabinovich, and B. A. Trakhtenbrot, in their Behavior Structures and Nets paper, uses the expression "partial multiset" without defining it. In context, they have an alphabet $\Sigma$, a finite multiset $A$ over $\Sigma$, and let $B \subset A$ be…
Clément
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map two sets of integers into two integers such that the number of intersections of these two sets can be found using only these two integers

Given two multisets $A$ and $B$, is it possible to map all of the elements of these two multisets into two integer (for example, integer $A_1$ will represent the elements of multiset $A$, and integer $B_1$ will represent the elements of multiset…
liam casey
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Can we apply the symmetric difference operation on two multisets?

I was just reading about multisets and operations on them. I didn't find mention of applying symmetric difference on multisets. Can we apply the operation on multisets like we can over sets? I decomposed the symmetric difference formula to the…
Kartikay Kaul
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Number of Elements of set of natural numbers = Number of elements of set having multiples of a number ?

What I mean to say is that " Is the Cardinal number of the set containing natural numbers same as the Cardinal number of the set containing the natural numbers divisible by 17 (or any other number ) " I feel like the answer should be true in a…
Kevin Peter
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What is the name for this Cartesian product-like operation?

I have two sets of multisets, like this: a: { { 11, 21, 31, 41 }, { 12, 22, 32, 42 }, { 13, 23, 33, 43 } } b: { { 21, 121, 131 }, { 22, 122, 132 } } I'm combining them together into another set of multisets by joining each multiset in a with each…
user198143
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Two sets given : solve $A \cup X = B$

Do you know how to solve this problem? I have two sets and need to solve $A \cup X = B. $ Thanks a lot for your help.
dzoni
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