Tagged Questions
1
vote
2answers
125 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
43 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
36 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
3answers
175 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 ...
3
votes
3answers
569 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
339 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.
...
0
votes
5answers
345 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 ...
1
vote
2answers
234 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 ...
2
votes
2answers
99 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
99 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
62 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 ...
2
votes
3answers
285 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 ...
