Sets Definition and 1000 Threads

Hey! :o I have to show that a set is recursive if and only if the set and its complement is recursively enumerable. I have done the following: $\Rightarrow$ Let $A$ the recursive set, so there is a Turing machine $M$ that decides the set $A$. We construct a TM $M'$ that semi-decides the set...

In the last year, I took a few Mooc online and I felt like my Physics was a bit rusty. So, I found myself a copy of Young and Freedman University Physics (13th edition) to do some self-study. The thing is, there is so many problems to do, I think that I’m on chapter 5 since the beginning of May...

Hi, So I was just going through my copy of The Emperor's New Mind, and I'm having a little difficulty accepting Godel's theorem , at least the way Penrose has presented it. If I'm not wrong, the theorem asserts that there exist certain mathematical statements within a formal axiomatic system...

Hi All, Say we have a finite collection ## S_1,...,S_n ## of sets , which are not all pairwise disjoint , and we want to find the minimal collection of the ## S_j ## whose union is ## \cup S_j ## . Is there any theorem, result to this effect? I would imagine that making the ## S_j##...

Homework Statement We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets...

I would appreciate any and all feedback regarding this document currently housed in Google docs. Basically, I generalize induction among natural numbers to an extreme in an environment regarding what I call grammatical systems. Then an induction principle is derived from that which holds in...

Homework Statement If I am given ##n(A)## and ##n(B)## for two sets A and B, and also provided with ##n(A\cap B)^2##. We are supposed to find ##n((AXB) \cap (BXA))##. Homework Equations My teacher said that the formula for ##n((AXB) \cap (BXA)) = n(A \cap B)^2##. I am not sure how do you get to...

Sets with cardinality ##2^{\aleph_0}##, that is, with cardinality of the set of real numbers, obviously have many applications in other branches of mathematics outside of pure set theory. For example, real any complex analysis is completely based on such sets. How about higher cardinality? Is...

Given the two standard definitions (1) A computable set is a set for which there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. (2) A computable number is a number which can be approximated to any degree of...

This is an elementary question that I may blush about later, but for now: given that a recursively enumerable set is a set modeling a Σ01 sentence, and a recursive set is a recursively enumerable set S whose complement ℕ\S is also recursively enumerable. Fine. But then, letting x→ = the...

Homework Statement Write formal descriptions of the following sets. (a) The set containing the numbers 1, 10, and 100 (b) The set containing all integers that are greater than 5 (c) The set containing all natural numbers that are less than 5 (d) The set containing the string aba (e) The set...

Homework Statement (I did not copy the problem statement, but basically solve the system of equations if there is solution and give a geometrical interpretation) P1: 2x - y + 6z = 7 P2: 3x + 4y + 3z = -8 P3: x - 2y - 4z = 9 Homework Equations Scalar triple product: n1⋅(n2 × n3) The Attempt...

Hello all, I probably should have posted this in a math forum but I don't know of any. Can anyone recommend a book/books on elementary number theory with exercises? My math background is not very strong with very little knowledge of set theory so it should be understood by me. We're covering...

Homework Statement Problem: Let A_1 , A_2 , . . . be a countable number of finite sets. Prove that the union S = ⋃_i A_i is countable. Solution: Included in the TheProblemAndSolution.jpg file. Homework Equations Set-theoretic algebra. The Attempt at a Solution Unless I missed something, it...

Hi I am new here! hopefully someone is kind enough to reply fast and help. so the question I am stuck is: Describe the relationship between two sets A and B ( A and B are non-empty) if: a. Pr(A|B)=Pr(A) b. Pr(A/B)=0 c. Pr(A/B)=Pr(A)/Pr(B) (Sorry guys can't get the fraction signs working! so...

Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...

Hi! Can someone please help me with this problem? I need to understand how to do it for my test tomorrow! I know this a spanning set for R^2, but the way I saw this problem solved was kind of lazy and not very helpful. S={(-1,4),(4,-1),(1,1)} I tried testing to see if it had just the trivial...

So I'm reading Naive Set Theory by Paul Halmos. He asks: His response is that no ##x## fails to meet the requirements, thus, all ##x##es do. He reasons that if it is not true for a given ##x## that ##x \in X~ \mathrm{for ~ every} ~X~ \mathrm{in} ~ \phi##, then there must exist an ##X## in...

Homework Statement Determine if they given set is a vector space using the indicated operations. Homework EquationsThe Attempt at a Solution Set {x: x E R} with operations x(+)y=xy and c(.)x=xc The (.) is the circle dot multiplication sign, and the (+) is the circle plus addition sign. I...

Homework Statement I know that the equation ##z = f(x,y)## gives a surface while ##w = f(x, y, z) ## gives an object that has the same surface shape on top as ##z = f(x,y)## but also includes everything below it. If these statements are correct, what is the level surface of a function of three...

Hey, In a course were we are treating phase transitions from a mathematically exact point of view, Cylinder sets were introduced. I'll first outline the context some more. So we are considering systems on a lattice with a finite state space for each lattice point, for simplicity. E.g. an Ising...

Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...

Hi there! I would like to understand the basic concepts of how mathematics is builded up from sets of elements to anything else (i.e: an equation) For example here is a formula: 2-1 So are there different sets of elements like A=(2) B=(-) C=(1) ? And maybe there is an operation with sets...

Homework Statement Ok I created this question to check my thinking. Are the following Sets: Open, Closed, Compact, Connected Note: Apologies for bad notation. S: [0,1)∪(1,2] V: [0,1)∩(1,2] Homework Equations S: [0,1)∪(1,2] V: [0,1)∩(1,2] The Attempt at a Solution S: [0,1)∪(1,2] Closed -...

Homework Statement I am trying to understand why ℕ the set of natural numbers is considered a Closed Set. 2. Relevant definition A Set S in Rm is closed iff its complement, Sc = Rm - S is open. The Attempt at a Solution I believe I understand why it is not an Open Set: Given that it...

Just a simple question regarding the nature of a compact set X in a metric space S: Does X necessarily have to be infinite? That is, are compact sets necessarily infinite? Peter***EDIT*** Although I am most unsure about this it appears to me that a finite set can be compact since the set $$A...

I am reading Tom Apostol's book: Mathematical Analysis (Second Edition). I am currently studying Chapter 4: Limits and Continuity. I am having trouble in fully understanding the proof of Bolzano's Theorem (Apostol Theorem 4.32). Bolzano's Theorem and its proof reads as follows...

Let's say you have 100 tickets of type A, and 100 tickets of type B in a box. Let's also say the probability to draw ticket A, for whatever reason, is twice that to draw ticket B. Is this problem, for all intents and purposes, mathematically equivalent to having 200 type A tickets and 100 type...

Hello! (Wave) Consider two sets $D=\{ d_1, d_2, \dots, d_n\}$ and $E=\{ e_1, e_2, \dots, e_m \}$ and consider an other variable $K \geq 0$. Show that we can answer in time $O((n+m) \lg (n+m))$ the following question: Is there is a pair of numbers $a,b$ where $a \in D, b \in E$ such that $|a-b|...

I am reading Walter Rudin's book, Principles of Mathematical Analysis. Currently I am studying Chapter 2:"Basic Topology". I have a second issue/problem with the proof of Theorem 2.43 concerning the uncountability of perfect sets in $$R^k$$. Rudin, Theorem 2.43 reads as follows:In the above...

I am reading Walter Rudin's book, Principles of Mathematical Analysis. Currently I am studying Chapter 2:"Basic Topology". I am concerned that I do not fully understand the proof of Theorem 2.43 concerning the uncountability of perfect sets in $$R^k$$. Rudin, Theorem 2.43 reads as follows...

I have two quick questions: With P being the power set, P(~A) = P(U) - P(A) and P(A-B) = P(A) - P(B) I'm told if it's true to prove it, and if false to give a counterexample. To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get...

Question: If \textbf{v}_1,...,\textbf{v}_4 are in \mathbb{R}^4 and \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3\} is linearly dependent, is \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3, \textbf{v}_4\} also linearly dependent? My Solution: http://s29.postimg.org/4wvwjlkqd/Linearly_Independent_Sets.png

Hello! (Smirk)Consider an implementation of disjoint sets with union, where there can be at most $k$ disjoint sets. The implementation uses a hash table $A[0.. \text{ max }-1]$ at which there are stored keys based on the method ordered double hashing. Let $h1$ and $h2$ be the primary and the...

Three sets $A,B,C$ given: (1)$A\bigcup C=B\bigcup C$ and (2)$A\bigcap C=B \bigcap C$ Prove: $A=B$

Homework Statement Hello, I'm not sure if it's the right place to post this exercise, but I'm learning it in a calculus course. I need to prove that: a) The complement of an open set is a closed. b) An open interval is a open set, a closed interval is a closed set. Homework Equations I have...

Hi! (Mmm) This: $$\subseteq_{\mathcal{P}A}=\{ <X,Y> \in (\mathcal{P}A)^2: X \subset Y\}$$ is a partial order of the power sets $\mathcal{P}A$. But, we have to take attention to the following fact: We cannot define an order over all the sets because if $R=\{ <X,Y>: X \subset Y \}$ is a set...

Homework Statement Let ##O## be a proper open subset of ##\mathbb{R}^d## (i.e.## O## is open, nonempty, and is not equal to ##\mathbb{R}^d##). For each ##n\in \mathbb{N}## let ##O_n=\big\{x\in O : d(x,O^c)>1/n\big\}## Prove that: (a) ##O_n## is open and ##O_n\subset O## for all ##n\in...

Hey everyone I just had a quick thought that was bothering me. For a set to be a basis it must be linearly independent and span the vector space. I've seen cases however of only two vectors forming a basis for R4 I don't see how two vectors could span 4 space or am I missing something. Thanks!

Hi! (Smile) We are given two sets $S_1$ and $S_2$. We consider that $S_1$ is implemented, using a sorted list, and $S_2$ is implemented, using a pre-order sorted tree. I have to write a pseudocode, that implements the operation Merge() of the sets $S_1$ and $S_2$. The time complexity of the...

Hello! (Smile) Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3)=\{ (t,t^2,t^3)/ t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3))=<Y-X^2,-X^3>$$ Finally, prove that the ideal $<Y-X^2,Z-X^3>$ is a prime ideal of $\mathbb{C}[X,Y,Z]$. Conclude that the algebraic set...

I am trying to understand the definition of compact sets (as given by Rudin) and am having a hard time with one issue. If a finite collection of open sets "covers" a set, then the set is said to be compact. The set of all reals is not compact. But we have for example: C1 = (-∞, 0) C2 = (0, +∞)...

Hi. I have a question about numbers of basis in quamutum mechanics space. Hamiltonian of harmonic oscillator is observable and have countably infinite sets |En>s Together with position or momentum basis identity equation is, |state>=\int|x><x|state>dx=\int|p><p|state>dp=\Sigma_n\...

Hello! (Wave) There is the following sentence in my notes: Let $A$ be a set. We define the set $I_A=\{ <a,a>, a \in A \}$. $$A \times A=\{ <a_1,a_2>: a_1 \in A \wedge a_2 \in A \}$$ Then $I_A$ is a relation, but does not come from a cartesian product of sets. Could you explain me the last...

Hello fine. I'm studying logic and great difficulties to understand its principles, and should prove some theories involving the laws of identity of sets of mathematics using the method of natural deduction, they are: a) A ∪ ∅ = A b) A ∩ ∅ = ∅ I am trying as follows, but I can not solve...

Homework Statement Hey guys, so this is a follow up from my previous post. So I have this Lagrangian for two coupled fields: \begin{split}...

I am kind of curious what topics to read to understand this concept more. Suppose I want to find a function f: A \rightarrow B, where if you look at the pre-image of any point b \in B, the size of the pre-image of b will be quite large. Essentially, I want to find functions that map from very...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences in Sections 2.1.14 and 2.1.15 B&K deal with ordered index sets in the context of direct sums and products of modules. Although it...

Hey! (Wave) Theorem (Russell's paradox is not a paradox in axiomatic set theory) The set of all sets does not exist. Proof We suppose that the set of all sets exist, let $V$. So, for each set $x$, $x \in V$. We define the type $\phi: \text{ a set does not belong to itself, so } x \notin x$...

I'm taking Multivariable Calculus at Rice. This is for fun, I'm not required to take it and I'm not a science major. Problem sets take me a really long time, I wonder if it's normal. We are using Vector Calculus by Marsden. The problems sets are like about 15 problems due every week. They take...