Questions tagged [multisets]

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

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45 views

Defining the Set of Rational Numbers

I was wondering about how to define the set of rational numbers, as I am currently learning about set theory in a class of mine. We were going through using set builders to define sets and produced ...
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How many groups of pentagonal flower bouquets can be formed?

A florist has three types of flowers: tulips, roses, and daisies. There are 4 tulips, 5 roses, and 6 daisies. These 15 flowers are to be arranged into three bouquets of 5 flowers each. Assume that ...
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How is Set Difference Defined for Multisets?

I'm interested in understanding the definition of set difference in multisets, but haven't been able to find a definition online. For an example, suppose $A = \{a, b, b, c, c, c\}$ and $B = \{a, b, c\}...
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1answer
30 views

Determine sets A and B

I need help solving this task, if anyone had a similar problem it would help me. The task is: Determine sets A and B if valid: $ A\cup B = \left\{x\in \mathbb{N} : x\le6 \right\}, A\cap B = \left\{x\...
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23 views

Some sum of two elements exceeds a third one in a finite multiset with bounded values

For a multiset $S = \{ a_1,a_2,a_3,\ldots,a_k \}$ where $k = 13$ and $1 \leqslant a < 32$, prove that there exists a subset $s = \{ a_i, a_j, a_k \}$ such that each sum of two elements exceeds the ...
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1answer
33 views

How do you translate 'positions of a value 2' into a mathematical notation when defining a bijection?

Let's say I have a multi-set: $(1,1,2)$ And I want to define a bijection from $A$ to $B$ such that: $(1,1,2)$ becomes {3} and, $(1,1,1)$ becomes {} and, $(2,2,2)$ becomes {1,2,3} or, in general, it is ...
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1answer
63 views

Find number of ways to divide a set into 2 parts

In how many ways can we divide a set into 2 parts having an element in equal number in both of resulting subsets. For example, multiset = {1, 2, 3, 5, 5, 5, 5} and ...
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1answer
133 views

Multiset equivalence

Let $a,b,c,d,e,f,g,h$ be natural numbers such that the multisets $\{a,b,c,d,a+b,c+d\}$ and $\{e,f,g,h,e+f,g+h\}$ are the same. Can we say that $\{a,b\}=\{e,f\}$ or $\{g,h\}$ ? and similarly $\{c,d\}=\{...
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65 views

Permutation with Repetition Index Conversion

I'm looking for the equation to determine the index of a permutation with repetition with known parameters. For example: A total of $9$ values, $4$ A's and $5$ B's Gives a total of $126$ permutations ...
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1answer
78 views

Counting k-combinations excluding duplicate mathematically

Let's say I have a multiset of integers a with a size n (here n = 10) $$a = \{1, 1, 2, 2, 3, 4, 5, 6, 6, 10\}$$ I'd like to know ...
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56 views

Number of combinations of two numbers from a list with repeating numbers? [duplicate]

I've tried googling it and looking it up on this website but since I don't know the technical term for this calculation I ran out of luck. Basically, if I have a collection of numbers (each of which ...
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37 views

Example of a function of a multiset where new elements reduce the result inversely proportionally to their size?

My math is rusty, but I'll do my best to make this legible: Let there be numbers $x, y$ where $x>y$. Let there be a finite multiset $A$ containing at least 1 element and $\exists z \in A$ where $z \...
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2answers
40 views

Notation for set union that results in a multiset

Is there a notation that replaces the "union" operator $A\cup B$ and emphasizes that the outcome should be considered a multi-set rather than a set? For example, if $A = \{1,2,3\}$ and $B =\{...
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2answers
48 views

generalization of multisets

Prove that for any $c,d \in \mathbb{R}$ and $k\in\mathbb{N}, \left({c+d\choose k}\right) = \sum_{j=0}^k \left({c\choose j}\right) \left({d\choose k-j}\right).$ I know how to show that ${a+b\choose k} ...
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1answer
31 views

Total number of combinations.

Total number of combination of $a,b,c$ and $d$ where elements can repeat up to maximum $m,n,o,p$ times respectively.(Note: order does not matter , i.e., ${a,b}$ and ${b,a}$ will be count as one only). ...
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1answer
84 views

Formula for the average of the lowest remaining value in a sequence of multisets

$B_1=(b_1,b_2,\ldots,b_n)$ is a finite multiset of positive real numbers $b_i$. To obtain $B_{i+1}$ we let $$m_i=\min\{B_i\}\quad\quad\quad\quad k_i=\min\{j\in\mathbb{N}:(B_i)_j=m_i\}\\ B_{i+1}=\left(\...
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2answers
31 views

Number of multisets

There's a question that asks how many $4$-element multisets are there whose elements are taken from the set $\{a, b, c\}$. I thought that there were $3$ options for each element, but they listed all ...
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21 views

How many permutations of the multiset $S = \{3\cdot a, 3\cdot b, 3\cdot c\}$ no two identical consecutive letters appear? [duplicate]

I have to find the number of permutations of $S = \{3\cdot a, 3\cdot b, 3\cdot c\}$ in which two equal consecutive letters do not appear. I'm trying to do it for inclusion-exclusion principle, but I ...
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1answer
53 views

Defining a multiset

Can the multiset $A=\{1,1,1,2,2,2,3,3,3,...,n,n,n\}$ be represented as $$A=\bigcup_{i=1}^{n}\{n,n,n\}$$ where $n$ is a positive integer. Or am I using the union notation completely incorrectly? If so, ...
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Cartesian product and relations of multisets and hybrid sets

I recently encountered multisets and hybrid sets (allowing negative multiplicities), and have a feeling they might be useful for something I'm trying to model. The definitions of both are clear to me, ...
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1answer
31 views

multisets in a subset

Define $B$ to be a multi-subset of a set $A$ if every element of $B$ is an element of $A$ and elements of $B$ need not be distinct. The ordering of elements in $B$ is not important. For example, if $...
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50 views

Confusion with mathematical objects (sets and multisets in particular)

The two sets $\{1,2,3\}$ and $\{1,1,2,3\}$ are equal, but the two multisets $<1,2,3>$ and $<1,1,2,3>$ are not equal. Multisets are generalizations of sets that allow repeated elements. A ...
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How do I prove sets are injective or surjective?

Let 𝑆 be the set of all lists (of any length) whose entries are only 0’s and 1’s, and let 𝑇 be the set of nonnegative integers. Define a function 𝐹: 𝑆 β†’ 𝑇 as follows: for any list 𝑠 in 𝑆, 𝐹(𝑠)...
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8 views

How to solve this equation with min function inside sum of multiset?

Let $m$ be a finite multiset of real numbers. Solve for $x\in R$ $$\frac{\sum_{i \in m}{min(x+1,i)}}{|m|} = x$$ In other words, find $x$ such that when you replace all numbers from $m$ that are ...
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1answer
34 views

Find elements of $\{0,1\}^4$

For a classroom repartition problem, I need to find a multiset of 8 vectors among $\{0,1\}^4$ such that Multiset: A vector can be present several times in the multiset Their sum with the regular ...
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18 views

Products of multisets tend toward mean

I need some hints on this... So I have multisets (sets where the same value can occur more than once) consisting of nonnegative integers $x$, with $0 \le x < p$. I would like to prove that if I ...
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20 views

Number of arrangements of multiset with different neighbours.

There are $n$ groups of objects with $x_i$ objects from group $i$ ($1 \le i \le n$). Compute the number of ways to arrange them in a line, such that there are no two consecutive objects from one group ...
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1answer
15 views

How many 10-combinations from the set $S$ with 5 distinct type $a,b,c,d,$ and $e$?

Problem. Given a multiset $S=\lbrace \infty.a, \infty.b, \infty.c, \infty.d, \infty.e \rbrace$, where $a,b,c,d$ and $e$ are distinct. How many the 10-combinations from $S$ where $a$ and $c$ at least ...
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59 views

What is the mathematical “set” function can be used to represent the uniqueness of a set?

What is the mathematical set function that can satisfy the following if X = (1, 2, 3, 4, 5), Y = (1, 3, 4, 2, 5), Z = (1, 1, 3, 2, 5); Then F(X) = F(Y) β‰  F(Z) what function "F" can be used to ...
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1answer
56 views

On day 1, Adam can take 1 walk, on day 2 he can take 2 (so on until day n), how many ways can he take 3 walks?

I'm trying to solve this question, but I'm not quite there and need some help. Question: Adam has just recovered from a serious leg injury and is encouraged to walk to aid his recovery. On day 1, he ...
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2answers
129 views

Unable to think about 10-combinations of a multiset

I am trying assignments of an Institute in which I don't study and i could not think about this problem. Problem is -> Determine the number of 10-combinations of multisets S= { 3.a, 4.b, 5.c} I ...
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21 views

Unordered sampling without replacement (inductive proof)

Question: I'm being asked to prove by induction that the size of the sample space equals the multiset equation: $\vert \Omega \vert =(\frac{m+n-1}{n})$ Specific Case There are M total balls ...
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4 views

CSR includes all $h_{ij}$

CSR includes all $h_{ij}$, where i union j includes all members of a set between 1 and n, OR needs to consist of all members of a set that contains multiples of each value in the set. Basically ...
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Enumeration of multisets satisfying a certain property.

Suppose $S$ is a infinite set and $R\subset S$ is also infinite. Now, we want to find the number of multisets $(M,\nu)$, with $M\subset S, |(M,\nu)|=n$, and having an additional property that there ...
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1answer
37 views

Combinatorics Question about Generating Functions and Multisets

I am faced with the following question from my undergraduate Combinatorics class: There are n aisles in a shop. We want to separate them into consecutive nonempty groups for different categories of ...
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116 views

Number of ways of making a sum of $k$ by choosing $n$ integers from a multiset

Problem Suppose that we are given a multiset of integers $A$ with the property that all elements in $A$ are between $a$ and $b$ (inclusive) where $a < b$. It is guaranteed that for all $i$ in $[a,...
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1answer
87 views

How many ways can you choose 12 items from 8 items types if you choose at least one of each type?

For example, how many ways can you choose $12$ items from $8$ item types? One possibility is that you choose every item from just one of the types. But what if we have the restriction that we must ...
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2answers
222 views

Category of multisets

I am trying to define a category $MSet$ of multisets as sets equipped with an equivalence relation. I will call such objects multisets. The other notion of multisets as pairs $(A,m_A\colon A\to\mathbb{...
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An equation involving multisets

For multisets $A, B, C, A', B', C'$, if $A \uplus B \uplus \{B \uplus C\} \uplus \{A \uplus \{C\}\}$ = $A' \uplus B' \uplus \{B' \uplus C'\} \uplus \{A' \uplus \{C'\}\}$, must $A=A',B=B',C=C'$, where $...
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46 views

Prove that Multiset and relation $\preccurlyeq$ is a lattice ($\preccurlyeq$ is defined like $\leq$)

Multiset is a set that can have more than one of each member for example $\{1,3,3,9\}$ is a Multiset. Let $\mathbb{K}$ be the set of all multisets that has exactly $k$ members. ($k$ is a fixed ...
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53 views

Denoting sets (permutations,tuples?),

I'll use and example to explain my question: Example: Let there be two distinct types $a$ and $b$ and each type has equal number of sets. There are two sets of type $a$: $\{1,2\}$ and $\{3,4\}$. ...
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26 views

Finding set with equal-size intersection given a set family

I'm wondering if there are any non-trivial sufficient conditions (or even just a google-search-friendly name for) for the following scenario: I am given a set family $\mathcal{F}$ on ground set $E$ (...
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2answers
18 views

Calculating r-combinations by hand (canceling out numbers in the denominator)

My textbook does an interesting cancellation process to simplify the r-combinations. How does this process work? How do you cancel out $4!$ with $19*18*17*16$? BTW how do you do this 31! with <...
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46 views

Category of multisets, spans and pullbacks

I want to define a category of multisets, $Multi$. To do this, I take the ambient category $SET$, and represent the multisets as functions. So, the objects $Multi$ are functions. We define ...
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Representing multisets

Multisets are like sets, but the "elements" can have multiplicities. An example is $M = \{ a,a, a, b,c,b,c \}$. We can present the multiset by giving the multiplicities for each set element. Can we ...
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2answers
56 views

Applying a function on any multiset of real numbers

I would like to define a function whose domain is any multiset of real numbers and image is a real number. To my understanding, the domain of a function that can be applied on any set of real numbers ...
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1answer
26 views

Reference request: Set of n-Multisets of elements in $\mathbb{N}$ is countable set

Let $n \in \mathbb{N}$ be fixed. I need a reference for the statement, that the collection of multisets of length $n$ with elements in $\mathbb{N}$ \begin{equation} M_{\mathbb{N}} = \{ \{a_1, ..., ...
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131 views

Permutations of pairs with restrictions

$n$ friends get together and decide to play a game. Thankfully one friend has a deck of $n$ cards, numbered from $1$ to $n$. How the Game Works The group splits into pairs (the game only works when ...
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Is possible to represent natural number X without 'splitting' it as additive-finitely repeated element set property?

I have a $Set$ of elements $n$ (a multiset). Each element of this set $n$ is the same, is a multiset because one element $n$ is replicable using addition operation, in other words we can have finitely ...
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1answer
505 views

Formula for r-Permutations of a Multiset

Suppose we have a multiset $M$, which contains $k$ distinct elements. Each element $x_i$ has multiplicity $n_i$ for each $i\in\Bbb{N}$ such that $0\le i<k$. $n$, the number of elements in $M$ ...

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