Set Definition and 1000 Threads

I am having difficulty with the following Exercise due next week. Prove that the set of all 2-element subsets of ##N## is denumerable. (Exercise 10.12 from Chartrand, Polimeni & Zhang's Mathematical Proofs: A Transition to Advanced Mathematics; 3rd ed.; pg. 262). My idea so far was...

I don't like Jech's textbook on set theory because he gives these definitions written in this bizarre language and he doesn't restate the definition in colloquial English. That mathematicians feel its unnecessary to give colloquial examples of their definition or examples, in my opinion, is a...

Homework Statement My textbook gives me this definition of a connected set. http://media.newschoolers.com/uploads/images/17/00/69/80/76/698076.png I have been working through my practice problems and I got to one that asked me to sketch the set given by|z+2-i|=2and note whether it is...

We define by recursion the set of sets {An:n∈ℕ} this way: A_0=∅ A_n+1=A_n ∪ {A_n}. I want to prove by induction that for all n∈ℕ, the set A_n has n elements and that A_n is transitive (i.e. if x∈y∈A_n, then x∈A_n). My thoughts: for n=0, A_1 = ∅∪ {∅} = {∅} then, for n+1: A_n+2...

Homework Statement Can you conclude that A = B if A and B are two sets with the same power set? Homework Equations The Attempt at a Solution I know intuitively that A and B have to be equal, because all the individual entities in the power set (you know what I mean) have to be in...

I'm trying to round out my math skills in order to apply to graduate school for physics and I've already taken all of the calculus offered along with linear algebra, power series etc... I'm wondering which would be better should I choose to take a math course this term: abstract algebra or set...

I'm an undergraduate studying math taking intermediate proof-writing courses, and there are certain basic identities of set theory and functions that still confuse me - i.e., I have to reprove them or think about them carefully every time. Examples: $$(A\times B)\cap (C\times D)=(A\cap C)\times...

Homework Statement Three tests are available to identify six different substances. Test 1 is true in the presence of substance 1,2,3, and 4 Test 2 is true in the presence of substance 3,4, and 5 Test 3 is true in the presence of substance 2,5, and 6 Using set notation to denote the...

Why do I run into trouble if I try to define the set of all groups? I get that defining the set of all sets could lead to paradoxes. But how is it that defining the set of all groups somehow leads to the same kind of problems? If I define the set of all groups as all the ordered pairs (x,y)...

Homework Statement Homework Equations A set is closed if it contains alll of its boundary points. A boundary point is a point where an open ball around that point has one point inside the ball that's in the set, and one point in the ball that's not in the set.The Attempt at a Solution As seen...

Please give me example of antipodal set in infinite dimensional?

Homework Statement In each of the two following open sentences P(x) and Q(x) over a domain S are given. Determine all ##x \in S## for which P(x) → Q(x) is a true statement. ## P(x): x \in [-1, 2]; Q(x): x^{2} \leq 2; S=[-1,1] ## Homework Equations According to truth values for →: a...

As a mathematician, what may you say are the beauties that you see in the Mandelbrot set??

I understand the definition of real numbers in set theory. We define the term "Dedekind-complete ordered field" and prove that all Dedekind-complete ordered fields are isomorphic. Then it makes sense to say that any of them can be thought of as "the" set of real numbers. We can prove that a...

I've been asked to fit the histogram with a Poisson distribution as part of a mostly independent learning thing. The data was produced through a stochastic simulation. Can someone get me started on how I would go about finding the expected distribution? If you need additional information...

Homework Statement Show that S is open if and only if ∀x ∈ S, ∃ a open ball B(x; r)(r > 0) such that B(x; r) ⊂ S And what we have is , let X be normed space, S ⊂X , Then S is close if and only if $$∀{x_{n}}⊂X, s.t. x_{n}->x \in X$$, x∈ S. A set S is open if and only if X\S is close...

Hey guys, I have just moved into a new place for my university studies. There are smoke detectors in the apartment(kitchen living room and bedroom), I never had those things in apartments that I use to live. I am quite worried about the one in the kitchen, because I heard that while cooking a...

So when we have an open cover of a set X means we have a collection of sets \{ E_\alpha\}_{\alpha \in I} such that X \subset \bigcup_{\alpha \in I} E_\alpha . My question comes from measure theory, on the question of finite \sigma -measures, The definition I'm readying says \mu is \sigma...

Homework Statement Determine the interior, the boundary and the closure of the set {z ε: Re(z2>1} Is the interior of the set path-connected? Homework Equations Re(z)=(z+z*)/2 The Attempt at a Solution Alright so z2=(x+iy)(x+iy)=x2+2ixy-y2 so Re(x2+2ixy-y2)= x2-y2 >1 So would...

Homework Statement Check if the following set of vectors are linearly dependent or independent: A) V1= \stackrel{1}{1} V2= \stackrel{1}{3} B) V1= \stackrel{\stackrel{1}{2}}{3} V2= \stackrel{\stackrel{2}{1}}{3} C) V1= \stackrel{1}{3} V2= \stackrel{2}{1} V3= \stackrel{-1}{2} Homework...

Find a countable set that is also open or prove that one cannot exist

Lemma: If A is an at most countable alphabet, then the set A^* of strings over A is countable. Proof begin: Let p_n be the n^{th} prime number: p_0 = 2, p_1=3, p_2=5, and so on. If A is finite, say A = {a_0, a_1, ... , a_n}, where a_0, a_1, ... , a_n are pairwise distinct, or if A is countable...

I know eigenvectors corresponding to different eigenvalues are linearly independent but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?

Homework Statement Homework Equations I have to use these set identities: The Attempt at a Solution Pretty sure this is impossible since it's an inequality.

Homework Statement Question #2. Homework Equations The Attempt at a Solution I've drawn a venn diagram for the left-hand side and the right-hand side and I can see that they're not equal but how do I provide a counter-example for this? Wouldn't a counter-example require an infinite number...

Homework Statement Homework Equations The Attempt at a Solution $$A-(A\cap B)=A-B\\ A\cap (A\cap B)^{ C }=A\cap B^{ C }\quad (set\quad difference\quad law)\\ A\cup [A\cap (A\cap B)^{ C }]=A\cup [A\cap B^{ C }]\quad (applied\quad A\cup \quad to\quad both\quad sides)\\ A=A\quad (absorption...

Homework Statement Homework Equations I have to use these set identities: The Attempt at a Solution Pretty sure this is impossible because there's no identity for the Cartesian product.

Homework Statement Let K \subset \mathbb{R^n} be compact and U an open subset containing K. Verify that there exists r > 0 such that B_r{u} \subset U for all u \in K . Homework Equations Every open cover of compact set has finite subcover. The Attempt at a Solution I tried...

Now i think i derived this correctly, but I'm not sure if it's correct, can anyone give me a confirmation? ##n-1/2=x## Where ##x## is the result of subtracting 1 from the observations and then dividing by 2. Then ##x+1=Median## Thank you :) Edit: Oops, this is the handicapped...

What is the precise definition of the open set? The definition I have been using until now has been that an open set is a set such that all of its points have some neighborhood that's contained in the set. The definition of neighborhood as far as I know is a collection of all the points within...

Hi Can you please help me with this problem? "What is the maximum size of a set A of logical expressions that only use →, p, q : each pair of elements of A are not equivalent?" I've found 6 different possible truth values. Is this the maximum size? If yes, how do I prove it? Thanks!

This is a concise question, so the title pretty much says it all. Also, this is not a HW question, but the idea has subtly popped up in two homework problems that I have done in the past. I cannot justify why the entire set would be bounded, because we know nothing of the nature of the...

Homework Statement -∏<arg(z)<∏ (z≠0) Homework Equations arg(z) is the angle from y=0The Attempt at a Solution Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put: -∏<arg(z)<∏ --> 0<arg(z)<2∏+k∏, (k ε Z) --> arg(z) \subset R --> arg(z) = R: all real numbers but I...

Find the value(s) of k such that f(x) is continuous everywhere: x^2-7 if x <= 2 and 4x^3-3kx+2 if x>2 Can you set the two equations equal to each other if only one of them has k in it?

hello How to you rigorously express the orthonormality of a complete set of eigenvectors (|q\rangle)_q of the position operator given that these are necessarily generalized eigenvectors (elements of the distribution space of a rigged hilbert space)? The usual unformal condition \langle...

Hey there! I will start of with saying I´m not very good at English when considering mathematical terms, neither an expert in Math. My question goes as this: I have a set of 1000 questions - which will be given in rounds with a set of 10. So every round, you get 10 questions out of...

Homework Statement I am not sure if set theory is precalc or not but here is my question. Find a pair set such that {a} belongs to the set and {a} is not a subset of S. The Attempt at a Solution So I thought that a set like this would work S = {{a}, b} because {a} belongs to the set...

Homework Statement Build the lagrangian of a set of N electric dipoles of mass m, length l and charge q. Find the equations of motion. Find the corresponding difference equations. Homework Equations Lagrange function L=T-V Lagrange's equations \frac{d}{dt}\left(\frac{\partial L}{\partial...

I'm at a bit of a loss how to do this. I suspect it's the set \left\{t_1^3, t_1t_2, t_2^2) \mid t \in {\mathbb C}\right\} . Certainly the polynomial x^2z^3 - y^6 vanishes on these points, but I'm not sure how to show the other inclusion. The only thing I can think of gets me close, but not...

Okay, so I am having a problem, I'm not the greatest at math, here is the problem: I need a set of reduction gears, my plan is to have a gear attached to a shaft (base diameter of this first gear is 33mm) with 50 teeth (maybe?) To turn a reduction gear that will be on another shaft that will...

In set theory a set is defined to be a collection of distinct objects (see http://en.wikipedia.org/wiki/Set_%28mathematics%29), i.e. we must have some way of distinguishing anyone element from a set, from any other element. Now a topological space is defined as a set X together with a...

The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder ##\mathbb{R}\times S^1##, torus ##S^1 \times S^1## etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set ##\left\{(x,y) \in...

If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.

I've seen open sets ##S## of a bigger set ##X## being defined as 1) for every ##x\in S## one can find an open disk ##D(x,\epsilon)## centered at ##x## of radius ##\epsilon## such that ##D## is entirely contained in ##S##. Where $$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$...

Homework Statement . I am trying to solve two exercises about complex sequences: 1) Let ##\alpha \in \mathbb C##, ##|\alpha|<1##. Which is the limit ##\lim_{n \to \infty} \alpha^n##?, do the same for the case ##|\alpha|>1##. 2) Let ##\mathcal M## be the set of the complex numbers ##c## such...

The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement...

Homework Statement 0<\left|x+3\right|<1/4 Homework Equations The Attempt at a Solution (-13/4)<x<(-11/4) and x\neq-3 Thanks in advance. This is my first post and I am unfamiliar with formatting this kind of stuff so I will work on getting better at that aspect.

Hey! :o I am looking at an exercise and I got stuck... $n\epsilon \mathbb{N},n>1$ $φ(n)=|\{1 \leq k \leq n :$ the greatest common divisor of $k$ and $n$ is $1\}|$ I am asked to find $φ(n)$,but I don't know how...

Hi everyone, I am working on a problem in Operations Research but I need to prove a property related to compactness of a set. Although I expect it is quite elementary, I have never studied Analysis at an advanced level so am not sure how to do it. I have an optimisation problem in which a...

I need some help finding an appropriate statistics model for some experimental data. Thanks in advance for any suggestions. I am trying to compare simulated results from a code that models nuclide concentrations in spent nuclear fuel to experimental data. These concentrations have...