Sets Definition and 1000 Threads

I'm not sure if I understood Vitali Sets correctly, so I just want to write what I understood (because I don't know if it's right): We have an equivalence relation where x \sim y \iff x-y \in Q. So if we look at the interval [0,1], each irrational number will have its own equivalence...

I'm familiarized with finding limits of most kinds of functions. I was struck by a problem: What if the variables of the function belong to different sets of numbers? My point being, given the function: f(n,q)=\frac{n}{q} With n belonging to the set of natural numbers and q belonging to the...

Homework Statement Question 1. Let U be a universal set, A and B two subsets of U. (1) Show that B ⊆ A ∪ (B ∩ A^c). (2) A = B if and only if there exists a subset X of U such that A ∪ X = B ∪ X and X\A^c = X\B^c. The Attempt at a Solution My attempt at a solution is as follows...

My question concerns F_\sigma subsets of \mathbb{R}. An F_\sigma set is one which can be expressed as a countable union of closed sets. I have several books that state that a countable intersection of F_\sigma sets need not be an F_\sigma set (indeed, such sets have their own designation...

I know that open intervals in R are homeomorphic to R. But does this extend to any dimension of Euclidean space? (Like an open 4-ball is it homeomorphic to R^4?) My book doesn't talk about anything general like that and only gives examples from R^2.

I was thinking about the following proposition that I think should be true, but I can't pove: Suppose that F is a group freely generated by a set U and that F is also generated by a set V with |U| = |V|. Then F is also freely generated by V. This is something that I intuitively think must...

Homework Statement If n(A - B) = 5, n(A' - B) = 4, n(A') = 10, n(B'-A') = 12. What is n(AUB) = ? Homework Equations The Attempt at a Solution I drew a big rectangle and inside 2 intersected diagrams A and B. I drew 5 dots in the (( of diagram A. Now that A' is complementary that...

Homework Statement A = {a,b,c} ; B U C = {c,d,e,f} ; (A∩B) U (C∩A) = ? Homework Equations The Attempt at a Solution A U B U C = {a,b,c,d,e,f} A ∩ {B U C } = {c} My answer: {c}

Homework Statement Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto its proper subset. Prove that every countable set is Dedekind infinite. The Attempt at a Solution I want to say that every countable set can be well ordered. I guess I could just pick some...

Consider the sequence $\{f_n\}$ of complex valued functions, where $f_n=tan(nz)$, $n=1,2,3\ldots$ and $z$ is in the upper half plane $Im(z)>0$. I want to show two facts about this sequence: 1) it's uniformly locally bounded: for every $z_0=x_0+iy_0$ in the upper half plane, ther exist...

hello I am struggled with a qustion let B be a countable subset of uncountable set A. Prove |A-B|=|A| i know how to prove that A-B is uncountable but how do i show 1:1 with A? thanks ahead guys

Hi, I'm reading through a proof of the existence of a nonmeasurable set. I've copied down the proof below more or less verbatim: In particular, I am trying to understand the significance of why ##\alpha## has to be an irrational number. Would the proof not hold if we used any other...

Homework Statement Use any method you wish to verify the following identity: (A \cap B) Δ C = ( A Δ C) Δ (A \ B) Homework Equations A Δ B = (A \ B) \cup (B \ A) = (A \cup B) \ (A \cap B) The Attempt at a Solution http://img17.imageshack.us/img17/48/14question14b.jpg I...

Posts: 10 I am a woodworker, and am designing a two part magnetic/spring lock for my blanket chest. The first part has 3 master buttons (primary buttons A, B, C), and the second part has 11 secondary buttons (1, 2, 3, ...11). What you do first is choose 1 of the 3 master buttons that opens...

Homework Statement Prove that if A,B, and C are nonempty sets such that A \subseteq B \subseteq C and |A|=|C|, then |A|=|B| The Attempt at a Solution Assume B \subset C and A \subset B (else A=B or B=C), and there must be a bijection f:A\rightarrowC...

Ok I understand the concept of infinite countability and that say the set of all rational #s is infinitely countable, but if I needed to represent the set how do I do that? S={xε rat. # : x= k , k ε a rational #}? that doesn't seem right. Also say I wanted to show a set of finite countable...

Hello all, while practicing set theory, I cam across this problem: If A and B are sets, prove that A x (B-C) = (AxB) - (BxC). This looks suspiciously like the distributive property but it's not. Is this simply a typo? Shouldn't the problem look like this: A x (B-C) = (AxB) - (AxC) Thanks...

Homework Statement To give some context, I'm trying to show that \mu(\bigcup^{\infty}_{k=1}A_{k})\leq \sum^{\infty}_{k=1}\mu(A_{k}) where μ is the Lebesgue measure and the A's are a countable set of Borel sets. Since the A's may not be disjoint, I'm trying to rewrite the left side of the...

This may seem like a silly question, but I'll ask it anyways. :) In the Munkres text, he proves this by showing that one-point sets are closed, which I completely understand why it follows that finite point sets are closed. He does so by showing that the arbitrary one-point set {x0} equals...

Let's say I'm looking at the infinite square well. Typically, given some arbitrary initial (normalized) wavefunction, we can decompose it into a linear combination of components of the complete set (on the interval [-a,a] or whatever) of sin's and cos's. Then, if you measure something like the...

Homework Statement Suppose that S and T are sets with outer content 0, prove that SUT also has outer content zero. Homework Equations C(S) denotes the outer content. C(S) = C(T) = 0 Also : C(S) = inf \left\{{ \sum_{k=0}^{n} A_k}\right\} where Ak is the area of one of the...

Pretty much every proof of this I've seen uses the axiom of countable choice at some part or another, and I never got why, since it's pretty cumbersome. Here's the sketch of a proof I wrote for the "fact" that a countable union of countable sets is countable: Let \ P:=\{\pi\in\mathbb{N}|\ \pi \...

I want to ask that, are you empty sets equal ?

Homework Statement A jet plane is flying at a constant altitude. At time t1=0 it has components of velocity vx=95m/s, vy=115m/s. At time t2=33s the components are vx=172m/s, vy=35m/s. Find average acceleration. Homework Equations avg acceleration=vfinal-vinitial/change in time The...

Homework Statement True or False: If S is a spanning set for a vector space V, then every vector v in V must be uniquely expressible as a linear combination of the vectors in S. Homework Equations The Attempt at a Solution For some reason, the answer to this question is false...

Hello, I need a help with the following: 1. Let $A$ be a transitive set, prove that $A\cup \{A \}$ is also transitive. 2. Show that for every natural $n$ there is a transitive set with $n$ elements.

Hello everyone! I want to show that all countable sets are closed. I can show that finite sets are closed, and the set of all natural numbers is closed by showing its complement to be a union of open sets. Now, can I start like this: A is a countable set. Every element in A can be "mapped" to...

A Formalization Musical "Sets" For those of you who have taken Music Theory IV (or upper division or even graduate courses on 20th Century Music Analysis), musical "set theory" should be a familiar concept. I use quotation marks because, as those who are familiar with mathematical set theory...

Homework Statement Label each set in the following Venn diagram as described in the attached pdf file. Homework Equations The Attempt at a Solution Unable to find a relationship that can describe the second set B. Obviously set A is a collection of Odd numbers.

For two sets X and Y let X^Y be the set of functions from Y to X. Prove that there is a bijection between (X x Y)^Z and X^Z x Y^Z. Attempt: I could not get any further from that "there must be a function S with S(f)=g and S(f')=g for any g, f' in X^Z x Y^Z, and where f is in (X x Y)^Z."

Attached are three figures. The problem in question essentially involves extracting data from a .CSV (MS Excel) file and .mdb (MS Access) file which I have done already but not correctly. Data set #1: gives me 'Wall Thickness vs. Time' ( [3400 - 8200] s , [0,1] in.) as shown in figure3.jpg...

Homework Statement Working in a banach space (X,\|\cdot\|) we have a sequence of compact sets A_k\subset X. Assume that there exist r_k>0 such that \sum_{k\in\mathbb{N}}r_k<\infty and for every k\in\mathbb{N}: $$A_{k+1}\subset\{x+u|x\in A_k,u\in X,\|u\|\leq r_k\}.$$Prove that the closure of...

The attached file illustrate the method I didnt understand this actually from lecture,and i try to search inside books but also didnt find any thing. The notes are brief so u can't get complete understanding So anyone could help me find relevant resources or such one example only to get the...

I'm trying to prove the homeomorphism between the open intervals of the real line and the open sets of the circle with the induced topology of R^2. Notice that the open sets of the circle is the intersection between the open balls in R^2 and the circle itself. Anyone can help me...

The solution attempt in the attacment

I have some confidence that this is the right idea. But whenever I have the slightest shred of doubt, I turn to the experts! :-pThus, before I write up my proof in my notes, does this look somewhat coherent? The problem states: “Determine whether the set of all functions from {0,1} to Z+ is...

Problem: -The Union of set A, set B and set C has 104 elements. -The Union of Set A and B has 51 elements -The Union of Set A and C has 84 elements -The Union of Set B and C has 97 elements -The Intersection of Set A and the Union of Set B and C has 17 elements. -Set C has twice as many...

Homework Statement prove by induction that if A and B are finite sets, A with n elements and B with m elements, then AxB has mn elements Homework Equations AxB is the Cartesian product. AxB={(a,b) such that a is an element of A and b is an element of B} The Attempt at a Solution...

Homework Statement So here is a "proof" from my measre theory class that I don't really understand. Be nice with me, this is the first time I am learning to "prove" things. Show that a connected open set Ω (\mathbb{R}^d, I suppose) is a countable union of open, disjoint rectangles if and...

Claim: If A is an inductive set of postive integers, then A is Z+. I tried to prove this two different ways for the fun of it. I would like to get some feedback concerning the correctness of both. Thank you. :-p Proof: By definition, Z+ is the intersection of all inductive subsets of ℝ. Since...

Good day! Im currently reading the book of Steven R. Lay's "Analysis with an Introduction to Proof, 3rd ed.". According to his book, if a subset S of ℝ contains all of its boundary then it is closed. But i find this wrong since if we consider S={xεQ;0≤x≤2}, then it can be shown that S...

Hello there, I am extremely new to mathematical analysis and do not have an idea how to prove the following questions. Could you please give me a hand and show me a way? Let At , t ∈ T, be a family of sets, and let X be a set. Prove the identities...

Homework Statement hopefully the writing is readable: http://i.imgur.com/VJ8vN.jpg All three if possible. Homework Equations none The Attempt at a Solution To be completely honest, I missed that whole week of lectures due to personal problems and I've had no chance to talk to an...

Hi Everyone, What are the difference between totally and partially ordered sets? Any examples would help except the fact that one holds reflexivity and another totality. Clarification of this would also be fine. Thank You

Prove that every closed set in $\mathbb{R}$ is the intersection of a countable collection of open sets. Let $G_n$ be a countable collection of open sets. Then we would have 2 cases either $x\in\bigcap G_n$ which is a point which is closed. Or we could $(a,b)$ in all $G_n$ but how to show that...

Here's what I want to do on the calculator. 1) Input sets of numbers, for example 3,6,9... in any notation, for example {3,6,9,infinity} or {x|x/3 >= 1 >= infinity} 2) use set theory (not now, but when I get into pre-cal/college.)

Homework Statement For a real number r, define A_{r}={r^{}2}, B_{r} as the closed interval [r-1,r+1], C_{r} as the interval (r,∞). For S = {1,2,4}, determine (a) \bigcup_{\alpha\in S} A{_\alpha} and \bigcap_{\alpha\in S} A{_\alpha} (b) \bigcup_{\alpha\in S} B{_\alpha} and...

Hi. I need some help with a proof. The question says: Let P be a probability function. Prove that for any finite collection of sets, the sequence A1,A2,...,An of pairwise disjoint sets, P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of Ai I think there must a mistake in question. My...

Hi, I'm not sure how else to phrase this.Let's say I have a linear transformation from R3 to R2. Let's assume in both spaces, I am using the standard topology with the standard euclidean distance metric. Does this mean that open sets in R3 will be mapped to open sets in R2 under this...

Homework Statement I'm using Introduction to Analysis 5th edition by Edward D. Gaughan. The question is: Prove (De Morgan) S\(\bigcap A_{\lambda}) = \cup(S\A) \lambda\epsilon \Lambda Where \Lambda A and S are sets (doesn't specify real or complex but assuming real) Homework...