Sets Definition and 1000 Threads

Is there a way in Mathematica to find the local maxima of a set of points? I have a fairly fine data set, and I can clearly see several peaks in it that I would like to know the numerical value of (as in, the highest point- I don't need a spline approximation or anything too fancy like that). I...

Homework Statement I'm self-studying Daniel Velleman's How to Prove It, and I'm wondering if there is some way to write that "Sets A and B are disjoint" using symbols, other than the A \cap B = \emptyset given in the book.Homework Equations The Attempt at a Solution I'm thinking that if A...

Let X be a non-empty set, and let S contain all countable subsets of X. Partially order S by inclusion. Let C be a totally ordered subset ("chain") of S, and let U = \cup_{E \in C} E It appears that U is not always countable: if it were, U would be an upper bound of the chain C, and U would...

I'm going through the set theory material in the appendix of Knapp's Basic Algebra. I want to make sure that I understand what he says is the set theoretic notion of the indexed cartesian product, \prod_{x\in S}A_{x}. He says that this can be thought of as the set of all functions...

Homework Statement Prove that if a given natural number M has no factors less than or equal to M1/2, then M is prime. Homework Equations None! The Attempt at a Solution So I was wondering, say I have two sets P and Q. Furthermore, say that P \bigcup Q = \mathbb{N} and P \bigcap Q =...

Homework Statement I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set U=\{x\in R^n : |x|\leq 1\}. While it is evident that...

Good evening, I was self-studying from a Discrete Mathematics book, and I ran across a question about infinite sets. Homework Statement The exercise asked to show that a set S is infinite if and only if there is a proper subset A of S such that there is a one-to-one correspondence between A...

Hi, I am stuck with the following proofs. In metric space here, A,B,C are subset of metric space (X,d) and C is bounded Problem 1.) d(A,B) <=d(A,C)+d(B,C)+diam(C) Problem 2.)|d(b,A)-d(c,A)| <= d(b,c) where 'b' belongs to 'B' and 'c' belongs to 'C'. Problem 3)- diam(A U B)<= diam A+...

Hi, I was wondering if anyone might want to help me with some less common dice probability. The dice mechanic is similar to the board game Risk (two sets of dice being compared), but the dice pools are varying numbers of 10-sided dice on either side (2 pools compared of a varying number)...

Is it true that finite sets don't have limit points?

I was reading an introductory chapter on probability related to sample spaces. It had a mention that for uncountably infinite sets, ie. in sets in which 1 to 1 mapping of its elements with positive integers is not possible, the number of subsets is not 2^n. I certainly find this very...

Hi, All: A simple question: If A is the sigma- algebra generated by a collection C of subsets of an ambient set X. Isn't it trivial that the sigma-algebra generated by A is A itself? One definition is that the sigma algebra generated by a collection S of subsets is the...

Homework Statement I'm having a little bit of trouble grasping the idea of null sets and power sets. Please check my answers to the following questions: Determine the cardinality: a. P(\oslash) b. P(P(\oslash)) c. P(P(P(\oslash)))Homework Equations n/a The Attempt at a Solution a. since...

How do I show the vectors, polynomials, and matrices generate the given sets? A subset of a vector space generates the vector space if the span of the subset is the vector space. The span is the set of all linear combinations. For 6, do I show the vector are linearly independent and can thus...

Homework Statement Incorrect Theorem: Suppose F and G are familes of sets. If \bigcupF and \bigcupG are disjoint, then so are F and G a) What's wrong with the following proof of the theorem? Proof. Suppose \bigcupF and \bigcupG are disjoint. Suppose F and G are not disjoint. Then we can...

Hello, I know one proof of this well known theorem that assumes on the metric of R being the standard metric. Does this result generalize to arbitrary metrics on R? thank you

Hi everyone, I came across a problem that requires knowing this fact. But can any open set in R^n be expressed as the countable union of "cubes". That is subsets of the form (a_1,b_1) \times ... \times (a_n, b_n) .

Basically there are 2 equations ; x+2y+3z = 1 2x+4y+6z=2 I put them into a matrix and row reduce to get 1 2 3 | 1 0 0 0 | 0 so we can say x = 1 - 2y -3z and let y and z = 0 to get a solution is (1,0,0) Now i need to find the nullspace to find the whole solution set; so x +...

Hi, I'm a second year physics undergrad currently revising quantum mechanics, and I came across a phrase about angular momentum which has confused me, so I was wondering if anyone could help. We looked at different components of angular momentum (in Cartesian) and decided that they did not...

In analysis we were shown the existence of non-Lebesgue measurable sets (eg a choice function over the rational equivalence partition of an interval). From the proof it seems that this means you can't assign number to the Lebesgue measure of this set i.e. if you say its measure is zero it's not...

I can add 2 dense sets together and get a non dense set right?

Before i say this i think transfinite numbers and infinite sets are really cool. But could you argue that infinite sets don't exist, I mean you couldn't show me one. I apologize if this is in the wrong section. I do believe in infinite sets but I was just wondering if you could argue this...

I have a dat file with multiple data sets, with the following structure: # t = 0.0 , ... -10.000 0.00001 1.000001 ... -9.000 0.00002 0.900001 ... ... 10.000 0.00005 1.000001 ... # t = 0.2 , ... -10.000 0.00301 1.000203 ... -9.000 0.02222 0.900043 ... ... 10.000 0.00025 1.000551...

Assume X is a metric space, then X and the empty set are both closed and open, am I correct?

Homework Statement Show that R \approx R+ , that is, the set of all real numbers is equivalent to the set of all positive real numbers Homework Equations The only relevant equation is finding one such that F:R\rightarrowR+ is a bijection. The Attempt at a Solution I've...

I feel like I have gone pretty far in math now, but I keep finding myself asking the same question. Say I had a series of data points from like a randomly collected survey or stock stock price graph over time etc. Is there a way to take this seemingly random and scattered data and turn it into...

Hello I'm having problems with actually proving this with some mathematics. Let A, B, C be sets. Prove that if A is a subset of B and B is a subset of C then A is a subset of C. Thanks.

Homework Statement Continuing with my Apostol efforts. From Section I 2.5: These exercises go over some of the absolute basics of sets. In 3. I'm given A = {1}, B = {1,2} and asked to decide whether some statements are true or false, proving the ones that are true. Seeing which ones are...

Hello, I have just been reading about the Zermelo-Frankel (ZF) axioms for set theory and thinking about their consequences. I understand that the Axiom of Regularity is needed in order to prevent contradictions like Russell's Paradox arising. That axiom says that any non-empty set A must contain...

Homework Statement show that if we define the following operation: let f=f(x) and g=g(x) be two functions in C[a,b] and define <f,g>=int(a to b) f(x)g(x)dx show that the conditions of therom are satisified with this operation. Use h=h(x) to help with part b this shows that this operation is...

So if we have a finite collection of disjoint non-empty sets, one can show using ZF only(with no need of AC) there is a choice function. I understand the reason for this. My confusion is when one goes to non-finite collection of sets. For example if the index set is the Natural numbers, why do...

I am trying to compare two groups (for statistical significance), a control and a treatment group across more than one time point, for a single variable. For example Control Treatment 0 sec x x 5 sec x x 10 sec x...

Let {X} be a set. Let {\mathcal{G}} be a non-empty collection of subsets of {X} such that {\mathcal{G}} is closed under finite intersections. Assume that there exists a sequence {X_h \in \mathcal{G}} such that {X = \cup_h X_h} . Let {\mathcal{M}} be the smallest collection of...

Homework Statement Suppose there exist three functions: f:A\stackrel{1-1}{\rightarrow}B g:B\stackrel{1-1}{\rightarrow}C h:C\stackrel{1-1}{\rightarrow}A Prove A\approxB\approxC Do not assume the functions map onto their codomains. Homework Equations The Attempt at a Solution I took a...

Homework Statement compute the first and follow sets of the following grammar S -> ACB | CbB | Ba A -> da | BC B -> g|λ C -> h|λ The Attempt at a Solution First(S) = First(ACB) U First(CbB) U First(Ba) First(ACB) = First(A) - {λ} U First(CB) First(CbB) = First(C) - {λ} U {b} First(B) =...

Hi! Can You please confirm that there is no contradiction between Cantor's theorem of countably infinite power sets and Galileo's paradox? - Thanks

Homework Statement True or false: Let S be any set in R2. The boundary of S is the set of points contained in S which are not in the interior of S. Homework Equations The Attempt at a Solution Common sense tells me true. I don't really understand it though, if S is an open set...

So I had this question in PF chat, but I decided this would be a better place for it. Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es. Is S larger than S', and if so, how much larger?

Homework Statement I am trying to prove that if f is continuous almost everywhere on [a,b], and if g is cont a.e. on [c,d], with f[a,b] contained in [c,d], then g composite f is cont. a.e. The Attempt at a Solution ------ Originally, my proof went something like this: f is cont...

Homework Statement Let U and V both have the same cardinality as R (the real numbers). Show that U\cupV also has the same cardinality as R. Homework Equations The Attempt at a Solution Because U and V both have the same cardinality as R, I that that this means \exists f: R\rightarrowU that is...

Homework Statement Is A = {0} union {1/n | n \in {1,2,3,...}} compact in R? Is B = (0,1] compact in R?Homework Equations Definition of compactness, and equivalent definitions for the space R.The Attempt at a Solution A is compact, but I can't seem to find a plausible proof of it... It should...

How to prove that set of real numbers is unbounded?

Homework Statement Prove by induction that the set [a_{n} | n_{0}\leq n \leq n_{1}] is bounded. a_{n} are the elements of the sequence (a_{n}) n \in N Homework Equations Definition of set bounded above: \forall x \in S, \exists M \in R such that x \leq M The Attempt at a Solution...

Homework Statement Convert (AC3)16 to base 10. I'm new to this kind of material. I would really appreciate your help for this one. Thanks, Roy Homework Equations The Attempt at a Solution

My problem is as follows: If we define d(A,B) = inf{ d(x,y) : x in A and y in B }, show that d(clos(A),clos(B)) = d(A,B), where clos(A) is the closure of A My attempt at a solution was this: Since A is a subset of the closure of A, then d(A,B) must be less than or equal to the distance...

What is the definition of a zero set and what exactly does it mean? I have come across different responses on the internet, but none of them explain really what it means or give good examples, I am having a rough time with this concept in real analysis. For example, how would I determine...

Homework Statement Show there's a G-delta set B, with E \subseteq B s.t. \lambda(E) = \lambda(B) Where \lambda is the Lebesgue measure and E is a Borel set. Homework Equations - G-delta set is a countable intersection of open set. - Lebesgue measure has properties: monotonicity...

A Union of a FINITE collection of closed sets is closed. But if it is an infinite collection? Can someone provide an example please?

I'm looking for advice. I have an assignment to train a neural network. I am confident in my abilities to write the code and train the neural network. The problems are: 1. I lack an idea. I'd like to do something interesting and potentially useful, obviously. 2. I need a rather large data...

If you take the intersection of non empty open subsets in Rn as n tends to infinity, such that U_1 \supseteq U_2 \supseteq U_3... Is it empty?