Set Definition and 1000 Threads

naive (intuitive) definition of "set" I happened upon a book by a Joseph Landin, once head of the math department at University of Chicago and subsequently Ohio State University, in which he gives this as a definition of a set and states this property: Shortly thereafter, he writes...

Express the set {X E R: (x+3) (7-x) ((x-2)^2) > 0} as a union of intervals

Here is the question: Here is a link to the question: Show that for any three sets A; B; C , we have: A - (B -C) = (A-B) U (A ? C)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Suppose that some infinite set S spans V. Then this means every vector in V is expressible as some linear combination of the vectors in S. Does this combination have to be finite? It couldn't be infinite, because that necessarily invokes notions of convergence and norm which do not...

I'm having some set theoretic qualms about the following argument for the following statement: Let V be a vector space of dimension n and let S be a generating set for V. Prove that some subset of S is a basis for V. The argument is as follows: If ##V = \{ 0 \} ## then it is trivial...

Let c_00 be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element x∈c_00, ∃N∈N such that xn=0,∀n≥n. Let {e_i}, i∈N be the set where e_i is the sequence in c_00 given by (e_i)_n =1 if n=i and (e_i)_n=0 if n≠i. Show that (e_i), i∈N is a basis for...

A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.29, P(A1 ∩ A2) = 0.07, P(A1 ∩ A3) = 0.09, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.02. The question is to find the...

Homework Statement Prove that if lim n→∞ (p_n ) = p in a metric space then the set of points {p,p_1,p_2, ...,} are closed. 2. Relevant information The definition of close in my book is "a set is closed if and only if its complementary is open." So I want to prove this by contradiction. I...

I am currently trying to understand linear independence and spanning sets. So the question that I have right now is, does a set of five linearly independent vectors always span F^5? Thanks for any help you can offer! Edit: Sorry for the confusion, I really had no idea how to begin this...

Homework Statement Find a set T of natural numbers such that 0 ∈ T, and whenever n ∈ T, then n + 3 ∈ T, but T ≠ S, where S is the set defined: Define the set S to be the smallest set contained in N and satisfying the following two properties: 1. 0 ∈ S, and 2. if n ∈ S, then n + 3 ∈ S...

Homework Statement Show that the set of \mathbb{R}^2 given by S = \{(x, y) \in \mathbb{R}^2 : x > y\} is open. Homework Equations The Attempt at a Solution Why is S the half plane that has boundary given by the line y = -x?

Homework Statement Find the volume obtained by rotating the regon boudned by the given curve about the specified axis Homework Equations The Attempt at a Solution y = secx, y = cosx, 0 <= x < = pi/3 This is what I set up. V = ∏∫ (secx +1 )^2 - (cosx +1)^2 dx I said R...

Homework Statement We have a metric space X=\cup X_k where X_k\subset X_{k+1} and each X_k is open. Show that for any Borel set E, there is an open set U such that \mu (U-E)<\epsilon . (Its supposed to be "U \ E".) Homework Equations \mu is a measure, so probably the important thing...

Homework Statement An interesting example of a ring: Begin with a nonempty set X and form the power set of X, P(X), which is the set of all subsets of X. On P(X), define addition + and multiplication × as follows: For A, B in P(X): A × B = A ∩ B A + B = (A\B) ∪ (B\A), where as...

Suppose we have a perfect set E\subset\mathbb{R}^k. Is there an open set I\subset E?

Homework Statement y = c1e3xcos(2x)+c2e3xsin(2x)+c3+c4x Homework Equations Differential Equations. The Attempt at a Solution I am having trouble what the roots are for the c3, and c4 parts. I know they are a repeated root, but is it just k= 0? It seems like it would work, but I...

Homework Statement Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. Homework Equations y = x^2 +1 , y = 9-x^2: about y = -1 The Attempt at a Solution I used disks. I said that r = 1 + ((9-x^2)-(x^2 +1 )) = 9 - 2x^2...

Hi, Using the definition of Hausdorff measure: http://en.wikipedia.org/wiki/Hausdorff_measure I am looking for a simple proof that Hd(C) is greater than 0, where C is the Cantor set and d=log(2)/log(3) Thank's in advance

Is 0 I am told. Is this an axiom, or can it be proven?

Hi, Say y=x2: the solution set is equal to the relation, which is also a set. What's the difference between saying "plot the relation y=x2" and "plot the solution set of y=x2"? Thanks for help.

My question concerns proving the set of non-negative integers of the form $a-dx ~~(a, d, x \in \mathbb{Z}, d \ge 1)$ is nonempty. This is the proof from my book. If $a \ge 0$, then $a = a-d\cdot 0 \in S$. If $a < 0$, let $x = -y$ where $y$ is a positive integer. Since $d$ is positive, we have...

Homework Statement Give S = {(x,|x|,2|x|) | x \in R} \bigcup {(0,2,4),(-1,3,6)}, find span(S) Homework Equations I know that span of a finite set of vectors is given by <a(0,2,4) + b(-1,3,6)+c(x,|x|,2|x|)>, where a,b,c are any real numbers. Can i use that same way to find the span of this...

How do you set a custom security level in Windows 8 to view Latex.

Homework Statement Prove that {m+n, m,n \inZ} is countable Homework Equations The Attempt at a Solution I Can prove it if I make a nxn scheme and put 1,-1,2,-2 along each side. This generates a table which when counted a long first,second etc. Diagonal hits all the numsers in the...

Homework Statement I want to show that the set $$ <1,x,x^2,\cdots ,x^n> $$ forms a basis of the space $$ P_{n} $$ where $$ P_{n} $$ contains all polynomial functions up to fixed degree n. The Attempt at a Solution I have already shown that the set $$ <1,x,x^2,\cdots ,x^n> $$ is linearly...

This proof makes no sense to me. The theorem to be proved is Theorem 44. {x,y} = {u,v} → (x = u & y = v) V (x = v & y = u) where {x,y} and {u,v} are sets with exactly two members, which can be either sets or individuals. The proof relies on: Theorem 43. z \in {x,y} z = x V z = y...

Tonight was an amazing night for stargazing so I drove about 10 minutes to a nice countryside road with barely any lights. It was really dark, and you could see a faint glow of the milky way. I had some apps to help me locate various galaxies and etc. and I wanted to see Andromeda but I...

I have an interesting 'What If' question. I understand that it was originally meant to be in jest, but I am interested to know what would happen. Here is the original joke: "There was a discussion in the math and physics departments of a university. It was about whether to allow calculators on...

1. Homework Statement . Let A and B be disjoint sets, A infinite and B enumerable. Prove that there exists a bijection between AuB and A. 3. The Attempt at a Solution . I have an idea of how to prove this statement, but I got stuck in the middle, so here is what I've done: There are just...

Ok, I don't think I'm on the right track here. I ASSUMED that the set of all countable collections \{I_k\}_{k = 1}^\infty of nonempty open, bounded intervals such that E \subseteq \bigcup_{k = 1}^\infty I_k is a countable set itself, which it probably isn't. I'm not even sure where to start...

Hi Say I have a finite data set (frequency, absorption) and I would like to find the corresponding dispersion. For this I could use the Kramers-Kronig (KK) relation on the absorption data. What I would do is to make a qubic spline and then perform the KK-transformation. However, the absorption...

of a countable collection of open intervals. I'm having a hard time seeing how this could be true. For instance, take the open set (0, 10). I'm having a hard time seeing how one could make this into a union of countable open intervals. For instance, (0,1) U (1, 10) or (0, 3) U (3, 6) U (6, 10)...

I realize that Russell's Paradox in naive set theory is considered to be, well... a paradoxical fallacy. Despite the fact that it is paradoxical and goes against logical intuition, is it really illogical though? It seems to me that the method in which the paradox arises is perfectly sound and as...

Homework Statement Define the w-limit set (omega) of a point. Show that w(x) is closed. Homework Equations The Attempt at a Solution The definition of a limit set is the set of points to which there exists a sequence t_n→∞ such that \phi(t_n,x) → y The second question. I was...

I'm reading Cantor's 1883 Grundlagen, it says a set is well-ordered if the set and it's subsets have first element, the next successor (unless it's an empty set or there is no successor). Note that the first element not neccessarily a least element. "Theory of sets" by E. Kamke also give the...

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This is a really dumb question that I have not found an answer for. I recently bought an erector set with a 3-6 volt motor. It has the option of 2 or 4 AA batteries. I am trying to figure out what the RPM's are with 4 AA batteries for a small hobby project of mine. I know this is simple...

Hi there! Is the following true? Suppose A is an open set and not closed. Cl(A) is closed and contains A, hence it contains at least one point not in A. If A is both open and closed it obviously does not hold.

\inHomework Statement Denote by d(x,A) = inf |x-y|,y \in A, the distance between a point x \in R^n and a set A \subseteq R^n. Show |d(x,A)-d(z,A)| \leq |x-z| In particular, x → d(x,A) is continuous Homework EquationsThe Attempt at a Solution I have no idea on how to prove this. I drew a...

According to a result of Paul Cohen in a mathematical model without the axiom of choice there exists an infinite set of real numbers without a countable subset. The proof that every infinite set has a countable subset (http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset) is...

Hello, I'm trying to set up an experiment to measure the distance between tracks in a CD, using only 3 items. I can only use a cardboard box, a ruler, and a laser pointer, and I will be using a double slit. So I'm thinking I will cut two slits in the carboard box, place it in front of the...

I'm working on monitoring and trending the performance of a memory controller. Without getting into the technical details, I'm trying to find a good way to visualize the data I've collected in a way that will be easy for me to update when I receive new data. There are about 450 individual...

Homework Statement Let S be infinite and, A a subset of S be finite. Prove that that the cardinality of S = the cardinality of S excluding the subset of A. Homework Equations The Attempt at a Solution We can write out the finite subset A as (x1, x2, ... xn) which can be put...

Homework Statement What is the cardinality of the set of all functions from N to {1,2}? Homework Equations The Attempt at a Solution I know the cardinality of the set of all functions coincides with the respective power set (I think) so 2^n where n is the size of the set. The...

I am supposed to prove: If A \neq \phi and B \neq \phi then A\times B \neq \phi The HINT in the back of the book gives: A \neq \phi \wedge B \neq \phi \Rightarrow \existsa\subseteqA \wedgeb\subseteq B so that (a,b) \subseteq A\times B I have 2 questions 1.Is it enough for the...

Hello Here is the problem statement. Let $X=Y = \{x\in \mathbb{R}\; :0<x<1\}$ . Define $ h\;:X\times Y\longrightarrow \mathbb{R}$ by $h(x,y)=2x+y$. For each $x\in X$, find $f(x) = \sup\{h(x,y)\; : y\in Y\}$. Here is my attempt. I let $S=\{h(x,y)\; : y\in Y\}$. I claim that $\sup S = 2x+1$. To...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Homework Statement This is taken from Spivak's Calculus Book Chapter 3 - Functions, Problem 9. Suppose ##f## is a function such that ##f(x)=1\text{ or }0## for each ##x##. Prove that there is a set ##A## such that ##f = C_A## ##C_A## is an indicator function, where ##C_A(x)=0## if ##x## is...

Hello all, I'm having a lot of trouble when it comes to set notation. For instance, what does (the set of all integers) $$Z^2$$ mean? What values are contained in this set?Sorry if I didn't use the MATH tags right.

Homework Statement A department consists of 5 men and 7 women.From this department you select a committee with 3 men and 2 women.In how many ways can you do this? Homework Equations Since the "overall set" (the entire department) is composed of both men and women and each has a specific...