# Questions tagged [set-partition]

This tag is for questions relating to "partition of a set" or, "set-partition", which is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in exactly one subset.

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### Let A be a 2n-element set. Find the number of pairings of A.

I am having trouble understanding how one of the solutions to this problem works: Let a pairing of A partition the set into 2-element subsets. Example: a pairing of {a, b, c, d, e, f, g, h} is {{a, b},...
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### Proof that the measure of a set is independent of its partition

I am struggling with writing and rigorousity and as an exercise from Terence Tao's Measure Theory book, I am trying to prove that the measure of a set E is independent of how it's partitioned. My ...
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### Intersection of interiors of sets in a partition of $\mathbb{R}^d$

Let $\mathcal{Q}=\{Q_1,...,Q_n\}$ and $\mathcal{P}=\{P_1,...,P_m\}$ be partitions of $\mathbb{R}^d$ with $n<m$. Assume all $Q_k\in\mathcal{Q}$ and $P_i\in\mathcal{P}$ are close and convex sets. ...
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### Number of ways of partitioning a union of $n$ sets

Suppose $E=A+B$ is the union of two sets $A$ and $B$.Then, we can have $4$ ways of partitioning $E$ into disjoint subsets: $\{AB,A∆B\},\{B, A\bar B\},\{A,\bar A B \},\{AB,A\bar B,\bar AB\}$ which is ...
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### General form of $\sigma$-algebra of countable set

Let $\Omega$ be a countable set. I want to show that every $\sigma$-algebra $\Sigma$ on $\Omega$ is of the form $$\{ \bigcup_{k \in K'}A_k : K' \subset K \},$$ where $A_k$...
1 vote
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### Does this relation partially order partial partitions?

By a partial partition of $X$, I mean a collection of non-empty subsets of $X$ that are disjoint. Given partial partitions $\mathcal{A}$ and $\mathcal{B}$ of $X$, define $\mathcal{A} \leq \mathcal{B}$ ...
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### Extension of equivalence relation concept

We have the definition of Cartesian Product as follows: Let $X$ and $Y$ be two sets, then the set $X \times Y = \{(a, b) : a \in X,~b \in Y\}$, is called Cartesian product of $X$ and $Y$ Elements of ...
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### Number of equivalence relations in a set of 5 elements

Let $X$ be a set s.t $|X|=5$ , How many equivalence relations are there from X to itself? I tried to do it "manually" and I tried to create all possible relationships, Considering $X$ has ...
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### Finding all set partitions where the elements in each part are restricted to specific subsets of the set.

I want to find the number of ways x people can pick different amount of items from a set, where we can somewhat pinpoint which particular items each person can pick. My case has 3 people picking up to ...
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### Do we have recurrences for evaluating the Partition Function on Graphs?

Inspired by this question about defining the partition function on non integers, I was thinking about what sorts of other objects can a partition function be defined on. I noticed that if we have an ...
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1 vote
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### Questions about the join and meet of partitions

In Aumann's Aggreeing to disagree" classic paper, the author defines for different partitions of a set $\Omega$, that refers to the state of the word, the join $\vee$ and the meet $\wedge$. For ...
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### Finer partitions mean more informative partitions? (I try to give an example and answer this question)

Is the following claim true? $\textbf{Claim:}$ The finer a partition $P$ it becomes, the more informative is. Let me give an example of a partition on some space where the Lebesgue measure is defined ...
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### Rationalization of $\frac{1}{\sum_{i=1}^{n}\sqrt{a_i}}$ (symmetric formula with multi-index)

Motivation No particular motivation, I'm just curious to see if an interesting result can emerge from such a seemingly simple problem. Problem Find the closed symmetric formula for the rationalization ...
1 vote
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### Bipartite graph definition in West's Introduction to graph theory

Why does it say "possibly empty" in the definition of bipartite graph? Wouldn't that mean that every graph is bipartite, moreover I don't think partitionins can have empty sets. Is this a ...
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### A Question Over set partitioning

There is a bit in the proof, the link, I couldn't understand. "There exists a sequence $(Π𝑛)$ of partitions, such that for each $𝑛$,$Π𝑛$ is an $𝑙+1/𝑛$-partition" How are we sure that ...
1 vote
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### Bijection between partitions of [n − 1] into k − 1 parts and partitions of [n] into k parts in which no part contains consecutive integers

Construct a bijection between unrestricted partitions of $[n − 1]$ into $k − 1$ parts and partitions of $[n]$ into $k$ parts in which no part contains consecutive integers. I was thinking of picking ...
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### Can someone help to clarify what " the data of an equivalence relation is the same as the data of a partition" means?

I am currently self-learning Modern Algebra on MIT OCW. I am at the lecture on cosets and am stuck on the theorem connecting equivalence relation and partition. I understand what the theorem implies, ...
Let $n$ be a positive integer, and let $p(n)$ be the number of partitions of $n$. For two partitions $p_1, p_2$ of the same integer $n$, we say that $p_2$ is a refinement of $p_1$ if the parts of $p_1$...