0
votes
1answer
9 views

Is every Partition a refinement of itself

Given a partition P on a set X, and seen as how every set is a subset of itself which means that every set in a partition is contained in itself does it follow that every partition contains itself as ...
1
vote
0answers
39 views

How to describe any partition a set

For ignore of a better word, I will use word "partition" try to describe what I mean. How to describe partition(where over lapping subsets are allowed) of a set mathematically? In another word, ...
3
votes
2answers
43 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
1
vote
1answer
34 views

How to talk about overlapping partitions of a set?

If I have a set $A$ and a number of sets $A_\sigma$ for each $\sigma \in \Sigma$ such that $$A = \bigcup_{\sigma \in \Sigma}A_\sigma$$ Is there a concise and eloquent way of saying this in plain ...
1
vote
1answer
47 views

proving antisymmetry of partition refinement

Suppose $P$ is the set of all partitions of some set $S$. $R$ is a binary relation on $P$, the refinement relation, defined as $(\Pi_1,\Pi_2) \in R $ if and only if for every $S_1 \in \Pi_1$, there ...
1
vote
2answers
44 views

Partitioning $\Bbb{N}$

Can we partition $\Bbb{N}$ into a finite union $$J_1 \sqcup J_2\ldots \sqcup J_N= \Bbb{N}$$ where $\sqcup$ denotes disjoint union. I'm guessing if we can then one of the $J_i$ must be infinite and ...
0
votes
1answer
36 views

Easy set partition problem

So I have this problem : Let A be a non empty set an P1 and P2 be two random partitions of the set A. Prove that the set is also a partition of A. I know that this is probably very easy to most of ...
0
votes
1answer
42 views

Invective function from $(0,1)$ to a partition

Consider the set $(0,1)$ and denote every $a \in (0,1)$ by it's decimal expansion $$ a=0.a_1a_2a_3\ldots $$ Now, define the equivalence relation $a \sim b$ if and only if $a_p = b_p$ for every prime ...
1
vote
3answers
112 views

Existence of uncountable set of uncountable disjoint subsets of uncountable set

"Can you to any uncountably infinite set $M$ find an uncountably infinite family $F$ consisting of pairwise disjoint uncountably infinite subsets of $M$?" Intuitively, I feel like it should be ...
0
votes
3answers
47 views

Powerset and Partition Creation

Let $A$ be a set with at least three elements If $\mathcal P = \{B_1 ,B_2, B_3\} $ is a partition of $A$, is $\{B_1^c , B_2^c,B_3^c\}$ a partion of A. So I am thinking the answer is no, since there ...
0
votes
1answer
48 views

Partitions on C = {i,-1,-i,1}

Let $ C = \{i, -1, -i, 1\}$ , where $ i^2 = -1 $. The relation $R$ on $C$ given by $xRy$ iff $xy = \pm 1$ is an equivalence relation on $C$. Give the partition of $C$ associated with $R$ I would ...
2
votes
0answers
53 views

Bell number with minimum bound on partition size

I know that the Bell number $B_n$ is the number of ways to partition a set of $n$ elements into distinct non-empty subsets. Is there a variant of this number that specifies the minimum number of ...
0
votes
3answers
85 views

Abstract Set Theory Question

can anyone explain what is going on here and how to solve this question please? Let $A$ be a nonempty set. Let $\{A_1,A_2\}$ be a partition of $A$. Consider the collection of set difference ...
2
votes
2answers
81 views

Finest and Coarsest Equivalences

According to a theorem in the Set Theory book I am reading, we can understand that equivalence relations partitons a set $X$ into distinct equivalence classes, $[x]$. I get that, but one of the ...
0
votes
1answer
135 views

How can I count the number of partitions of S with exactly n parts?

If I have a set $S$ of $n$ elements, is there a way to find the number of partitions of that set with $k$ "parts/cells"? For example, if set $S = \{a, b, c, d\}$, there are 15 total partitions of ...
2
votes
4answers
124 views

Number of size 1 partitions of the empty set

hDisclaimer: This is a homework problem, but I'm just asking for clarification, not a solution. We're asked to prove $S(0,1) = 1$, where $S(n,k)$ is "the number of different partitions of [a set of ...
2
votes
3answers
291 views

Prove that any partition induces a unique equivalence relation.

Given any partition $D$ of $A$, $\exists !$ equivalence relation on $A$ from which it is derived. Can someone please help me solve this problem? thanks.
1
vote
1answer
50 views

Partitioning a set

I have this question: Is this collection of subsets a partition on the set of bit strings of length 8: The set of bit strings that end with 111, the set of bit strings that end with 011, ...
-1
votes
1answer
70 views

Prove that this set (involving fractional part of any rational number) is a partition of the set of rationals.

For any rational number $x$, we can writte $x=q+\,n/m$ where $q$ is an integer and $0\le n/m<1$. Call $n/m$ the fractional part of $x$. For each rational $r\in \{x : 0\le x<1\}$ ,let $A_r = \{ ...
1
vote
2answers
357 views

common knowledge and concept of coarsening partition

Here is a proof of the equivalence between my definition and Aumann's for "common knowledge". I'm assuming some familiarity with set partitions. Aumann's definition is in terms of the ...
3
votes
1answer
72 views

Finite Partitions of the Unit Interval

Does the unit interval have a finite partition $P$ such that no element of $P$ contains an open interval? I would think that the answer is no, because each element of $P$ would have Lebesgue measure ...
0
votes
1answer
51 views

Is this a valid partition?

Say we have the $S$ which is the set of all compositions of n >= 0 with an odd num. parts. Define $S_1$ to be the set of all compositions of n >= 0 with an odd num. of parts where at least one part ...
1
vote
4answers
971 views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
6
votes
3answers
2k views

How many ways to merge N companies into one big company: Bell or Catalan?

There's a famous interview question variously credited to Microsoft, Google and Yahoo: Suppose you have given N companies, and we want to eventually merge them into one big company. How many ...
3
votes
1answer
458 views

Number of partitions of a set with size constraints

One of my students made some experiments on partitions of sets. He found some results and I asked him, if he can prove some statements. After two weeks he had no result, so maybe one of you can help. ...
2
votes
5answers
644 views

General questions about equivalence classes and partitions

1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with ...
2
votes
2answers
375 views

Proof of a Proposition on Partitions and Equivalence Classes

I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like, Proposition: If $\{A_i|i\in I\}$ is a partition of ...
1
vote
2answers
331 views

Proving a relation between 2 sets as antisymmetric

Let $U = \{1,...,n\}$ And let $A$ and $B$ be partitions of the set $U$ such that: $\bigcup A = \bigcup B = U$ and $|A|=s, |B|=t$ Let's define a relation between the sets $A$ and $B$ as follows: $B ...
3
votes
2answers
128 views

a formal name of this partition

Consider a set $A=\{1,2,3,4,5\}$, is there any terminology for the following partitions of $A$ ? (1) $A=\{ \{1\},\{2\},\{3\},\{4\},\{5\} \}$ (2) $A=\{\{1,2,3,4,5\}\}$.
1
vote
2answers
140 views

Partition of a Nonempty Set $X$

Let $X$ be a nonempty set, and $\{A_\alpha : \alpha\in I\}$ be a partition of $X$. If $B\subseteq X$ such that $A_\alpha\cap B\neq\emptyset$ for every $\alpha\in I$, is $\{A_\alpha\cap B : \alpha\in ...
1
vote
2answers
87 views

Verifying That A Collection Is A Partition of $\mathbb R^3$

Can someone possibly explain this to me, I'm having difficulties visualizing it: For each $r\in\mathbb{R}$, let $A_r=\{(x, y, z)\in\mathbb{R}^3 : x+y+z=r\}$. How can I tell if this is a partition of ...
3
votes
3answers
504 views

“Converting” equivalence relations to partitions

There is a direct relationship between equivalence relations and partitions. Is there a way to simply use an equivalence relation's definition to get the matching partition? And what about the other ...