Set Definition and 1000 Threads

The Godel theorem shows that the standard Peano axiomatization or arithmetic is undecidable. However, there is an alternative Presburger's axiomatization of arithmetic, which is decidable. Similarly, the standard ZFC axiomatization of set theory is undecidable. For instance, the continuum...

Homework Statement The carbon dioxide molecule can be considered a linear molecule with a central carbon atom, bound to two oxygen atoms with a pair of identical springs in opposing directions. Study the longitudinal motion of the molecule. If three coordinates are used, one of the normal...

Homework Statement Find a spanning set for ##P_4##. Find a minimal spanning set. Use Theorem 2.7 to show no other spanning set has fewer elements. Would simply like someone to check my answers as the book I'm using did not provide a solutions manual. Thank you. Homework Equations Theorem 2.7...

Homework Statement [/B] Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is *Bold characters are vectors* x_1= 1 + 3t x_2 = 2 - t x_3 = 3 + 2t where t is a parameter and can be any number. a) How many pivots are in the row echelon form of A? b) Let u, v, w be the...

Here is a Science mag news story about several studies using a large, internationally gathered, set of brain scans (MRIs). The large numbers of data sets allows for statistically meaningful numbers of cases with similar problems, diseases, or genetics to be compared. Some unexpected findings...

I have been writing this science fiction novel in the form of a blog so that I can get feedback. It's based on the direction that technology is progressing on a whole, but in a "distant" future when we basically Ray Kurzweil got his way, but it didn't turn out exactly how he predicted. Chapter...

Homework Statement Suppose that { |ψ1>, |ψ2>,...,|ψn>} is an orthonormal basis set and all of the basis vectors are eigenvectors of the operator Q with Q|ψj> = qj|ψj> for all j = 1...n. A particle is in the state |Φ>. Show that for this particle the expectation value of <Q> is ∑j=1nqj |<Φ|...

Homework Statement Find the complement C' of the set C with respect to the space U if: 1. U = {(x,y,z) : x^2+y^2+z^2 ≤1}, C = {(x,y,z) : x^2+y^2+z^2 = 1} 2. U = {(x,y) : |x| + |y| ≤ 2}, C = {(x,y) : x^2 + y^2 < 2} Homework Equations Definition of complement: The elements of the space that...

Suppose I have an infinitely thin foil that absorbs 1% of incoming radiation. It therefore emits 0.5% of the incoming radiation in both directions perpendicular to the plane of the foil. That is, transmission is 0.5%. How to calculate transmission for a series of such foils? There will be a lot...

Hi all, at the moment I am doing my Master Thesis and have the following problem. I am trying to measure Data and asign it a proper timestamp. My problem is, that the data is coming in bursts and the timestamps I assign with the software are wrong. The controller I am using for monitoring the...

I am not sure whether this is posted before but it looks really good to me. http://www.tcm.phy.cam.ac.uk/~bds10/aqp.html

Hello I have a question regarding something we wrote in class today. Let ##A## be a bounded subset of ##R^n##, let ##f,g:A\to \mathbb{R}## be integrable functions on A. ##a)## if ## A## has a volume and ##\forall x \in A :m\leq f(x) \leq M## then ##mV(A)\leq \int_{A}f(x)\leq MV(A)## this...

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Lemma 1.2.10 ... Duistermaat and Kolk"s proof of Lemma 1.2.10 (including D&K's definition of a...

So I made this diagram of the order imposed on the set of topologies on a 3 point set. Each small circle is a topology with the circle elements being the open sets. The larger circles join isomorphic topologies together. The ordering was messy when I tried to draw inclusion from each topology to...

Homework Statement X is the set of all integers n that satisfy the inequality 2 <= |n| <=5 What are the elements of set X? Homework Equations Provided above. The Attempt at a Solution In my view X should have: 2, 3, 4, 5 elements. I can't understand how X have negative values. Some body...

Hey! :o Let $V$ be a $\mathbb{R}$-subspace with basis $B=\{v_1 ,v_2, \ldots , v_n\}$ and $\overline{v}\in V$, $\overline{v}\neq 0$. I have shown that if we exchange $\overline{v}$ with an element $v_i\in B$ we get again a basis. How can we show, using this fact, that the intersection of all...

I am going over some of my notes, and I cannot understand unions, here is the selection I am having trouble with How does the union of four different sets equal just one of the sets? Should the union of 4 sets be the four different sets instead of one. I am missing something fundamental to...

I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ...

Homework Statement You're in Paris, working on your first novel and short on cash. The owner of the cabaret downstairs has offered to take care of your rent if you'll help her design a climate control system for the place. It gets too hot and damp in the summer, and too cold and dry in the...

What do MHB members think are the best books at an undergraduate or senior undergraduate level on set theory ... Further what are the best books on set theory at a graduate level ... ... Peter

The mirror that I'm talking of, is situated in Hazarduari Palace in Murshidabad, West Bengal, India. This place was once the capital of Bengal under the Nawab before the British defeated the Nawab in the historic battle of Plassey, 1757. The history is not that important here, but the mirror...

I am trying to find the most efficient way to select points on a 2d plane from a set that maximizes the area of the of the shape they define when joined together. The points are all paired (sharing the same A->B vector), with these pairs also appearing mirrored about the origin. Here is an...

I'm trying to figure out something regarding the span of a set of vectors. Say we have a set of vectors ##V=\{v_1, v_2, ... v_k\}## in ℝn. If k < n then the set does not span ℝn. Why is this? Are there vectors in ℝn that aren't combinations of the vectors in ##V##?

Homework Statement Control systems subject. Have a DC motor setup on a difference op-amp, then with gain, and the axel moves a wiper on a pot used for feedback. I understand the concepts reasonably well enough to complete the assignment. I'm using MATLAB scripts to approximate parameters...

Homework Statement Prove that {##x \epsilon \mathbb{R} : x^2 \ge 1##} is "not" bounded below. EDIT: I Looked closely and realized there is a "not" that we all had to write in...sorry if you lost some time.. Homework Equations Defintion: We say a nonempty subset ##A## of ##\mathbb{R}## is...

Homework Statement 1. You are consulting for a screenwriter who is working on a science-fiction movie about the trials and tribulations of an space-faring family. She wants the script to be reasonably realistic from a scientific perspective, so she is seeking your advice on a variety of scenes...

Hi All, Let ##s: \{s_1,s_2,...,s_j \}## be a collection of elements contained in the sets ##S:=\{S_1,S_2,...S_k \}## , no relation between ##j,k##; a given ##s_i## may be contained in one or more ##S_n##. I want to find a minimal "cover" for ##s##, i.e., the smallest subcollection of sets in...

I am puzzled by the derivation of the Cantor set. If the iteration of removing the middle-thirds leaves an uncountable set of points, it seems the iteration had to be performed an uncountably infinite number of times. Is this the case? If so, that seems paradoxical to me.

My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory. Let me describe a method to define the membership relation...

Homework Statement Let ##E## be a nonmeasurable set of finite outer measure. Show that there is a ##G_\delta## set ##G## that contains ##E## for which ##m^*(E)=m^*(G)##, while ##m^*(G-E) > 0##. Homework Equations ##E## is a measurable set if and only if there is a ##G_\delta## set ##G##...

Homework Statement Willy leaves point A and Milly leaves point B, both of them are heading toward each other's starting point. They both have constant speeds, but Milly is faster, and they pass each other at noon without stopping. Milly arrives at point A at 1:00 pm, and Willy arrives at point...

Hello! I was introduced in the Real Analysis class the notion of ordered set. I am not sure I understand the concept of linear and non-linear ordering. Can someone explain this to me? Thank you!

Homework Statement Suppose that pn → p and each pn lies in lim S. We claim that p ∈ lim S. Since pn is a limit of S there is a sequence (pn,k)k∈N in S that converges to pn as k →∞. Thus there exists qn = pn,k(n)∈ S such that d(pn, qn) <1/n. Then, as n→∞ we have d(p, qn) ≤ d(p, pn) + d(pn, qn) →...

Homework Statement Suppose that S is a closed set. We claim that Sc is open. Take any p ∈ Sc. If there fails to exist an r > 0 such that d(p, q) < r ⇒ q ∈ Sc then for each r = 1/n with n = 1, 2, . . . there exists a point pn ∈ S such that d(p, pn) < 1/n. This sequence in S converges to p ∈ Sc...

How much would time pass between watching the sun set from ground level and then watching it set again from the top of a sky scraper? I heard once that this could be done using one of the towers of the World Trade Center. So I assume one could also do this using the Sears / Willis Tower in...

Homework Statement Assume ##A## and ##B## are given sets, and show that there exists a set ##C## such that ##C = \{\{x\} \times B : x \in A\}##. Homework EquationsThe Attempt at a Solution To show that this set exists, I thought to maybe use the subset axioms (or the axiom schema of restricted...

Homework Statement Prove that if ##A \subseteq B##, then ##\bigcup A \subseteq \bigcup B##. Homework EquationsThe Attempt at a Solution This is a simple problem, but I just want to make sure I am writing out the proof correctly: Suppose that ##A \subseteq B##. We want to show that ##\bigcup A...

Homework Statement Say you're bowling and you roll the ball down the lane. The bowling ball has just knocked the first bowling pin forward and that pin's forward velocity is now much larger than the bowling ball's velocity before their impact. How can the pin travel forward faster than the ball...

Homework Statement Assume that ##x## and ##y## are members of a set ##B##. Show that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## Homework EquationsThe Attempt at a Solution I know that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## iff ##\{ \{x\}, \{x,y\} \} \subseteq...

I need a clear-cut explanation of vacuous truth, as I can't seem to wrap my head around it. I guess this more precisely comes down to why we organize the truth table for the conditional statement the way that we do. Also, in connection to this, I'm wondering why the empty set is a subset of all...

Homework Statement Show that ##\bigcup \{\mathcal{P} X : X \in A \} \subseteq \mathcal{P} \bigcup A## Homework EquationsThe Attempt at a Solution Suppose that ##c \in \bigcup \{\mathcal{P} X : X \in A \}##. Then by definition this means that ##\exists a \in A## such that ##c \in \mathcal{P}...

Homework Statement In a physics lab, 0.30 kg puck A, moving at 5.0 m/s [W], undergoes a collision with 0.40 kg puck B, which is initially at rest. Puck A moves off at 4.2 m/s [W 30 degrees N]. Find the final velocity of puck B. Homework EquationsThe Attempt at a Solution One thing I haven't...

Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...

Homework Statement Let ##E## be a closed set of real numbers and ##f## a real-valued function that is defined and continuous on ##E##. Show that there exists a function ##g## defined and continuous on all of ##\Bbb{R}## such that ##f(x) = g(x)## for each ##x \in E##. Homework EquationsThe...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (c) ... ... Exercise 2.1.10 Part (c) reads as follows:I am unable to make a meaningful start on...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (b) ... ... Exercise 2.1.10 Part (b) reads as follows: I am unable to make a meaningful start on...

I plan on building a swing set for my daughter, and I always loved the feel of a good tall swing. I like the long range of motion, the nice glide ... but is there such thing as too tall? I was planning on going about 14' off the ground with the cross beam .. then I figured, well, let's go with...

Ordering on the Set of Real Numbers ... Sohrab, Exercise 2.1.10 (a) ... I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (a) ... ... Exercise 2.1.10...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.12 Part (1) ... ... Exercise 2.1.12 Part (1) reads as follows: I am unable to make a meaningful start on...

Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.12 Part (1) ... ... Exercise 2.1.12 Part (1) reads as follows: I am unable to make a...