Sets Definition and 1000 Threads

I have two sets of deterministic numbers, collected in the two vectors: x=[x(1),...,x(n)] and y=[y(1),...,y(n)]. My (determinstic) theory says that x(i)=y(i) for all i=1,...,n. But instead, I want to assume that the numbers x and y are stochastic. If we let f(.) be a pdf, does this mean that I...

Homework Statement While studying a book "analysis on manifolds" by munkres, I see a definition of measure zero. That is, Let A be a subset of R^{n}. We say A has measure zero in R^{n} if for every ε>0, there is a covering Q_{1},Q_{2},... of A by countably many rectangles...

Homework Statement Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero. Homework Equations A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')}...

Hi, I hope someone can help me out with this problem: Let set S be defined by (x in En :f(x) <=c} f: En -> E1 is convex and differentiable and gradient of f(xo) is not 0 when f(x) = c. Let xo be a point such that f(xo) = c and let d = gradient at xo. Let lamda be any positive number and...

Okay, so I'm struggling with understanding where I went wrong. The instructor feels like I don't understand the material and when she presented my explanation to a colleague, he too agreed with her. I would really appreciate if someone could tell me the first part of where I went wrong in my...

Homework Statement The following is all the information needed: Homework Equations There are, of course, all the basic rules of logic and set identities to be considered. The Attempt at a Solution Not really sure how to attempt this one, to be honest. I know that (A ⊆ B) can...

I don't get the meaning of "connected" in the chapter of planar sets. The textbook said " An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S" So do i just randomly pick 2 points in S to check if they are both in...

There are quite a number of threads (and FAQ's) that discuss why the speed of light is the "number" that it is, but I'm having difficulty finding some information on what is causing, or setting, the limit. So, let me ask and answer a different question as an example of the what I am tryinh to...

Homework Statement Let S1 = {a} be a set consisting of just one element and let S2 = {b, c} be a set consisting of two elements. Show that S1 × Z is bijective to S2 × Z. Homework Equations The Attempt at a Solution So I usually prove bijectivity by showing that two sets are...

Homework Statement Determine whether the following statements are true or false a) Every pairwise disjoint family of open subsets of ℝ is countable. b) Every pairwise disjoint family of closed subsets of ℝ is countable. Homework Equations part (a) is true. we can find 1-1...

Homework Statement Let C be a Cantor set and let x in C be given prove that a) Every neighborhood of x contains points in C, different from x. b) Every neighborhood of x contains points not in C Homework Equations How can I start to prove? The Attempt...

Homework Statement Find a condition on a metric space (X,d) that ensures that there exist subsets A,B of X with A\subset B such that diam(A)=diam(B).Homework Equations diam(A)=\sup\{d(r,s):r,s\in A\}; A\subseteq B\implies diam(A)\leq diam(B).The Attempt at a Solution Well I know examples of...

Homework Statement Suppose (X,d) is a metric space, and suppose that A,B\subseteq X. Show that dist(A,B)=dist(cl(A),cl(B)). Homework Equations cl(A)=\partial A\cup A. dist(A,B)=\inf \{d(a,b):a\in A,b\in B\}The Attempt at a Solution Its clear that dist(cl(A),cl(B))\leq...

Counting Lists With Repetition Homework Statement How many ways can you create an 8 letter password using A - Z where at most 1 letter repeats? Homework Equations The Attempt at a Solution I'm not sure how to attack this problem but first I thought that A-Z considers 26 letters...

Hello everyone, I was wondering if I could get some advice for the following problem: I have two algebraic sets X, X', i.e. X = V(J), Y = V(J'), and let I(X),I(Y) be the ideals of these sets, i.e. I(X) ={x \in X | f(x) = 0 for all x \in X}. I am trying to show that I(X \cap Y) is not always...

Homework Statement For the sets A and B, prove that A \cap B \subseteq A \subseteq A \cup B The Attempt at a Solution I am guessing I should look at only two of them first? A \subseteq A \cup B What conditions do I need?

Homework Statement Show that every compact set must be closed. I am looking for a simple proof. This is supposed to be Intro Analysis proof. Relevant equations Any compact set must be bounded. The Attempt at a Solution Suppose A is not closed, so let a be an accumulation...

Homework Statement Suppose A = {1,{1},{1,{1}}} Then is {{{1}}} an element of A? The Attempt at a Solution I am thinking A has the elements are only 1, {1}, {1, {1}} But {{{1}}} has only the element {{1}} While A has the element {1,{1}}, you can't just take out the...

Homework Statement Let M be a metric space, A a subset of M, x a point in M. Define the metric of x to A by d(x,A) = inf d(x,y), y in A For \epsilon>0, define the sets D(A,\epsilon) = {x in M : d(x,A)<\epsilon} N(A,\epsilon) = {x in M: d(x,A)\leq\epsilon} Show that A is...

For the infinite cardinal numbers (of which there are infinitely many), do they each necessarily correspond to some set? I mean we know that aleph-naught corresponds to N, c (aleph-one by continuum hypothesis) corresponds to R, but what about all the other infinite cardinals? Is it possible...

I was wondering, is S a subset of S-bar its closure? For example, if p belongs to S, does p belong to S-bar too? Does it go the other way, S-bar is a subset of S? If it is true that S is a subset of S-bar does this automatically mean that S is closed? Thanks

Is it true in general that if f is Lebesgue integrable in a measure space (X,\mathcal M,\mu) with \mu a positive measure, \mu(X) = 1, and E \in \mathcal M satisfies \mu(E) = 0, then \int_E f d\mu = 0 This is one of those things I "knew" to be true yesterday, and the day before, and the...

Homework Statement Prove: a set in a topological space is closed and nowhere dense if and only if it is the boundary of an open set.Homework Equations Basic definitions of closed, nowhere dense, open and boundary.The Attempt at a Solution One direction is easy. Let A \subset X be a subset in...

Homework Statement Consider the set in E^2 of points {(x,y)|(x,y)=(1/n,1-1/n), where n is a positive integar}. Find the limit points, interior points and boundary points. Determine whether this set is open or closed. Homework Equations The Attempt at a Solution I figured, 0,1 must...

Homework Statement Consider the following LOP K: Max z = 4x_1 + 3x_2 s.t 7x_1 + 6x_2 \leq 42 2x_1 - 3x_2 \geq -6 x_1 \geq 2, x_1 \leq 5, x_2 \geq \frac{1}{2} (a) Decide whether the solution sets are feasible i) x = (3.5, 2.5)^t ii) x = (2.1, 4)^t iii) x = (5,1/2)^t (b) Graph the...

Can you give some examples of the infinite sets that are uncountable and that are not continuous? I know the infinite sets that are countable and discrete, and I know the continuous sets, but couldn't find an example for the above situation.

A theorem of real analysis states that any open set in \Re^{n} can be written as the countable union of nonoverlapping intervals, where "interval" means a parallelopiped in \Re^{n}, and nonoverlapping means the interiors of the intervals are disjoint. Well, what about something as simple as an...

Homework Statement I'm having some major trouble this these two questions. Sketch the set, s, where s = {z| | z^2 - 1 | < 1 } ... z is a complex number Sketch the set, s, where s = {Z| | Z | > 2 | Z - 1 | } ... Z is a complex number 2. The attempt at a solution This is supposed to...

I have to decide whether the following is true or false: If A1\supseteqA2\supseteqA3\supseteq...are all sets containing an infinite number of elements, then the intersection of those sets is infinite as well. I think I found a counterexample but I'm not sure the correct notation. I to...

Homework Statement Ai and Bi are indexed families of sets. Prove that Ui (Ai \bigcap Bi) \subseteq (UiAi) \bigcap (UiBi). Homework Equations The Attempt at a Solution Suppose arbitrary x. Let x \in {x l \foralli\inI(x\inAi\bigcapBi) This means x \in{x l...

The book I am reading says that \bigcap \phi because every x belongs to A \in \phi(since there is no such A ) , so \bigcap S would have to be the set of all sets. now my question is why every x belongs to A \in \phi.In other word I don't completely understand what this statement mean. sorry if...

Homework Statement Hi, someone could help to draw the forces or the velocities in matlab to check if they are properly calculated?. Homework Equations I am adding the forces like that phi0 = phi0+dt.*force; so I do not know how to get the velocity. but I would like to get the...

Hi If I have measured the resonance frequency of three sets of resonators and calculated the mean, variance and standard deviation for each set. How do I add the three variances and standard deviations to get an overall variance and standard deviation? Well, I know that the standard...

Homework Statement Decide if the following represents a true statement about the nature of sets. If it does not, present a specific example that shows where the statement does not hold: If A_{1}\supseteqA_{2}\supseteqA_{3}\supseteqA_{4}\supseteq...A_{n} are all sets containing an...

This is the question: Let A be an open set and B a closed set. If B ⊂ A, prove that A \ B is open. If A ⊂ B, prove that B \ A is closed. Right before this we have a theorem stated as below: In R^d, (a) the union of an arbitrary collection of open sets is open; (b) the intersection of any...

Homework Statement Let I denote the interval [0,\infty). For each r \in I, define Ar = {(x,y) \in RxR : x2+y2 = r2}, Br = {(x,y) \in RxR : x2+y2 \leq r2}, Cr = {(x,y) \in RxR : x2+y2 > r2} a) Determine \bigcupr\inIAr and \bigcapr\inIAr b) Determine \bigcupr\inIBr and...

Homework Statement I'm working on a proof for Intro. to Algebra, and the problem deals with proving that the operation of Symmetric Difference imposes group structure on the power set 2^x (set of all subsets) of a given set X. There is a part of my proof that I'd like to get some advice on...

Homework Statement What are the truth sets of the following statements? List a few elements of the truth set if you can. c) x is a real number and 5\in{y\inR|x^{2}+y^{2}<50} The Attempt at a Solution I believe this says 5 is a member of the set of possible values for y, while y is...

I have been consulting different sources of analysis notes. My confusion comes from these two definitions \begin{defn} Let S be a non-empty subset of $\mathbb{R}$. \begin{enumerate} \item $S$ is Bounded above $ \Longleftrightarrow\exists\,M > 0$ s.t. $\forall\, x\in S$, $x\leq M$...

"Concrete" non-measurable sets I've had Vitali's proof of the existence of non-(Lebesgue) measurable sets branded into the side of my brain over the years. However, the proof always critically relies on evoking the axiom of choice. Has anybody every demonstrated a non-AoC construction of a...

Homework Statement All the b's in f[b] should be capitalized for the problem statement and attempt; I had it in the latex but it showed up lower case in the post I don't know why, my apologies =p. If f:X \mapsto Y and A \subset X, B \subset X, is: (a) f[A \cap B] = f[A] \cap f[B] in...

Let Q be the group of rational numbers with respect to addition. We define a relation R on Q via aRb if and only if a − b is an even integer. Prove that this is an equivalence relation. I am very stumped with this and would welcome any help Thank you

Homework Statement Let A be a set of real numbers that is bounded above and let B be a subset of real numbers such that A (intersect) B is non-empty. Show that sup (A(intersect)B) <= sup A The Attempt at a Solution I don't know how to start but tried this... Let C = A (intersect) B So...

Hi, Is there a characterization of subsets of the Cantor space C that are closed but not open? As a totally-disconnected set/space, C has a basis of clopen sets; but I'm just curious of what the closed non-open sets are.

Hi people, Let U(N) be the unitary matrices group of a positive integer N . Then, U(N) can be viewed as a subspace of \mathbb{R}^{2N^2} . I am curious what the open sets of U(N) are in this case. If it has an inherited topology from GL(N,\mathbb{C}) , what are the open sets of...

I have a 5000x1 vector and am trying to write a function to calculate an answer for entry 1-11, then 12-22, then 23-33, etc. ... I've been trying to use a 'for' loop, basically: for i = (??) x=i+1 end Not sure what to put in the ? area. I want it to spit out answers for each set of...

Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?

Suppose I have 2 sets of data: day1 and day2. I want to fit a model to both data sets and then compare them to each other to see which one fits the model the best (the fit is done with a computer using non-linear least squares method). The RMS of the fit would be fine except that day1 has much...

Hello. As I understand, in the classical logic it's impossible to "take", for example, the set of all sets. I was wondering: is it possible to create a logic where that is possible by changing some of the basic postulates by which logic works? Or is it impossible for all logics? Thank you.

Given matrices in a vectorspace, how do you go about determining if they are independent or not? Since elements in a given vectorspace (like matrices) are vector elements of the space, I think we'd solve this the same way as we've solved for vectors in R1 -- c1u1 + c2u2 + c3u3 = 0. But I'm...