**242 sets questions.**

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference. A symmetric difference of sets A and B is the set (A \ B) ∪ (B \ A).
I know that ...

The Problem:
I am currently analyzing a simple program that takes a file of length $n$, splits it into its individual words (seperated by white space) and adds those words to a set:
...

I understand that if $A \leq B$ and $A$ is not a recursively enumerable set then $B$ is also not a recursively enumerable set.
(and if $A \leq B$ and $B$ is decidable then $A$ is decidable)
What if ...

I was wondering what is the most efficient algorithm to solve something like the following:
You have $P$ people.
You have $T$ tasks, each of which is a set of sets that represent all of the possible ...

I have the following problem :
Given an Array of integers such as : V = {9,4,1,2,4,3} and a number such as N = 12
divide into K parts, so each part adds N or less, taking the best solution.
I ...

I have a list of sets $L$. I want to partition the sets in $L$ to produce a new list $L'$ that is a Laminar Set Family
Concretely:
For any $L'_i, L'_j \in L'$ if $L'_i \not\subseteq L'_j$ and $L'_j ...

I have a set, and i want to find all its subsets. what is the best time complexity to find it?
What is the most efficient algorithm to find all subsets of a set?

I have the following problem. The problem can be formulated in three different ways
Given sets $B_{-n},\ldots,B_n \subset \{1,\ldots,m\}$.
Find $i,j \in \{-n,\ldots,n\}$ with $|i| \neq |j|$ and $i,...

Let there be a set of cardinality $n\in \mathbb{N}$. Let there also be $n$ subsets of that set. What is the smallest k such that union of some $n-k$ of those subsets is of cardinality at most $k$? The ...

Given a collection of sets $S_i$ of disjoint subsets $sub_i$ of a set $X$,
find a set $A$ of disjoint subsets $asub$ such that each one of these subsets is subset or equal to at most one subset in ...

I am trying to understand the larger problem of the decidability of the equality of two DFAs. I understand that this problem can be solved using minimizing DFAs, but my textbook states this can be ...

Let $n$ be any positibe integer and set $N=\{1,\dots,n\}$. Now select two arbitrary but different subsets of $N$, say $S$ and $S'$. We are interested in finding a function $\pi(A)=\sum_{i\in A}a_i$ ...

Question-
Prove: Every decidable set is Turing reducible to the empty set.
Can anyone help me with this please?
All reductions tutorials I've seen use practical examples of reduction such as sipser'...

Let $S$ be a set (say positive integers $\leq$ N) and $f$ an involution ($f$ is bijective, $f\cdot f=id$, e.g. xor with a constant). $g$ is a idempotent mapping choosing an arbitrary representative ...

I've been back and forth about this one. I have the following theoretical homework problem, which describes the SET-INTERSECTION problem. In my homework, it's ...

Given an integer $k$ and $n$ sets $A_1,\ldots,A_n$, denote $U=A_1\cup A_2\cup\cdots\cup A_n$, $A_i^0=A_i$ and $A_i^1=U\backslash A_i$. The problem asks whether there exists $(b_1,\ldots, b_n)\in\{0,1\}...

Background
I have a list of 50 million $A-A_i$ pairs, where $i>1$, and $A$ and $A_i$ are some text. I need to extract the $A$ values from the list, so I get a new list of unique $A$ values.:
$$
\...

I have a set of (integer) ranges and want to compute the (possibly non-disjoint) set of all subsets of overlapping ranges. The data structure used for the output is not of particular importance to me; ...

I have been studying data structures. In that I have come across topics like Array being defined as Power set of cross product of set of objects and set of natural number and list being defined as ...

In this question, I asked what the difference is between set and type. These answers have been really clarifying (e.g. @AndrejBauer), so in my thirst for knowledge, I submit to the temptation of ...

Let $A\subseteq \{1\ldots n\}$ with $|A|=\alpha n, 0<\alpha\leq1$. Now we start generating random sets $B_i \subseteq \{1\ldots n\}$ with $|B_i|=\beta n$ where $0<\beta\leq\alpha$.
How many $...

Let $m, n \in \mathbb{N}$ and $n \le m$. Given $k$ subsets $X_1, X_2, \dots, X_k$ of $\{ 1, 2, \dots, m \}$ and $k$ nonnegative integers $a_1, a_2, \dots, a_k$, find all subsets $Y$ of $\{ 1, 2, \dots,...

I was digging through research articles to find a data structure that solves the dynamic sorted dictionary problem (representing any subset $S$ of a universe $U = \{0, \ldots, u\}$ with member/...

I am given an list of numbers and A number-s. I need to find out the collection(s) of numbers from the list of numbers whose sum corresponds to the given number s.
...

For homework I have the task
Assuming P ≠ NP, is the following set NP-complete:
{(G,w) | G is a Graph and w is a Hamilton cycle in G}
and I don't understand how to show that a set is NP-complete.
I ...

This question concerns the time complexity of outputting the unions of subsets of a given set.
Given $m$ subsets of an $k$-element set, can the union of those sets be computed in linear time with ...

I have the following problem:
Given a set a of n positive integers, write a backtracking C function that prints out all the subsets of a such that the product of their elements is p. Use an array ...

We have given a multiset of $N$ integer, both positive or negative. Consider all $2^N$ subsets, sorted by their sum (the empty subset has sum 0). We want an algorithm that outputs only the first $K$ ...

EDIT: I've now asked a similar question about the difference between categories and sets.
Every time I read about type theory (which admittedly is rather informal), I can't really understand how it ...

I'm interested to read about type theory, but I'm quite a beginner. I know what sets are and how to work with them, but I don't have a deep understanding of set theory. I don't really understand the ...

I am confronted with the following problem:
Let S be the family of all m-subsets of $[n] = [2m]$
let $S_1, S_2 \in S$ be distinct sets and let the state of storage be $State_1$ after stream $S_1$ is ...

The principle (called a Löwenheim–Skolem theorem by Huth and Ryan) states
Let $\phi$ be a sentence of predicate logic such that for any natural
number $n \geq 1$, there is a model of $\phi$ with ...

We can denote by $X\to X$ the set of all functions from $X$ to $X$.
Therefore, we can use the following statement to say that $f$ is a function from $X$ to $X$:
$$f\in X\to X$$
But we usually state ...

so for a homework assignment i need to prove the following:
We have arbitrary languages L1⊆∑1*, L2⊆∑2*, L3⊆∑3*, L4⊆∑4*Prove that the followging is either true or ...

I have a large number of sets, A, B, C, ... where each set includes a few integers. I would like to find the set that includes the highest number of other sets.
A ...

This was an interview question that I was told is supposed to be an open ended discussion of the trade-offs.
You have a collection of comparable objects and want to be able to do the following:
1. ...

Can a Turing machine $M_A$ determine if the Turing machine $M_B$ accepts the set $W_k$?
I am curious about the answer to this as I am thinking about using the truth value of it on using it for a ...

Let's consider finite grid of points with size of $N$ by $M$ and set of $x$ points ($x$ is small number, up to 10, $N$ and $M$ are big numbers, up to 30000 )). Each of the $x$ points is described with ...

I have a collection of objects, with certain properties (let say 3 - zone, type, owner) only having a small predetermined possible set of values (like enum).
This is just a simple (javascript) array ...

I have a list of nodes l = [1, 2, 3, ... , n] and a list of tuples p = [(1, 2), (2, 3), ...], where the latter represents which ...

I'm reviewing my background in Algorithms and DS design. Specifically I never went through the van Emde Boas Tree. Though I can undestand the proto-vEB with related picture. I'm struggling to ...

Let $A$ denote a set that contains a relatively large number of different strings. Let $S_i$ denote these strings.
Let $B$ denote a set of sets such that each subset contains a (relatively small, ...

Given two sets of items
$A = { a_1, .., a_N }, B = { b_1, .., b_M },$
and assuming a connection weight $w{_i}_j \ge 0$ between any possible pair $(a_i, b_j)$ that contains one item of each set, how ...

In a partially ordered set L, an antichain is a subset A of L such that no two elements of A are comparable.
Antichains are commonly used to represent upward-closed subsets of L, that is, sets S such ...

I have two large sets of integers $A$ and $B$. Each set has about a million entries, and each entry is a positive integer that is at most 10 digits long.
What is the best algorithm to compute $A\...

Consider a sequence $s$ of $n$ integers (let's ignore the specifics of their representation and just suppose we have can read, write and compare them in O(1) time with arbitrary positions). What's ...

Is the intersection of infinitely many recursive sets $\bigcap_{i}U_{i}$ (where each set is different ) recursive? Recursively enumerable?
I know the union need not be recursive, because deciding if ...

I am trying to make sure my intuition for the following question from an assignment is correct
Prove or disrove: if $G = (V, E)$ is a graph and $I_1$ and $I_2$ are independent sets in $G$, then $I_1 \...

I am trying to think of how to optimize the following problem:
Let $S = \{1,2,...,N\}$. How many ways can $S$ be partitioned into non-empty subsets $P_1$ and $P_2$ such that $sum(P_1) = sum(P_2)$? I ...

I´m looking for an efficient algorithm that will find reverse cartesian products.
Mathematically, given $S \subseteq T^n$, I want to express $S$ as a union of sets $A_{i,1} \times A_{i,2} \times \...

- algorithms
- data-structures
- optimization
- combinatorics
- complexity-theory
- terminology
- formal-languages
- computability
- graphs
- partitions
- time-complexity
- np-complete
- graph-theory
- algorithm-analysis
- reference-request
- trees
- strings
- regular-languages
- type-theory
- space-complexity
- discrete-mathematics
- coding-theory
- enumeration
- finite-sets
- logic

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