Sets Definition and 1000 Threads

Let ##I## denote the interval ## [0, \infty )## . For each r ## \in I ## define: ##A_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} = r^{2} \}## ##B_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} \leq r^{2} \}## ##C_{r} = \{ ## ... ## : ... > r^{2} \} ## a.)...

I'm writing the paper on this experiment I just did. Basically I took sets of data for two variables (x,y) and I fit the points to a line in Origin to extract the value that I was trying to measure (J). *Using generic variables here* I found a value for J where J = [8*∏*x*a(b+c)] / y I...

Suppose A and B are open sets in a topological Hausdorff space X.Suppose A intersection B is an empty set. Can we prove that A intersection with closure of B is also empty? Is "Hausdorff" condition necessary for that? Please help.

Homework Statement Let A,B, and C be subsets of universal set U. Prove the following A. If U=A union B and intersection of A and B is not an empty set, then A= U\B B. A\(B intersection C) = (A\B) union (A\C) Homework Equations no relevant equations required The Attempt at a...

Homework Statement So if a question asks you for a pair set, with some criteria, is it enough to just say S = {a,b} or do you need something extra? Also this is another question that is from my set theory class, if the question defines a function and to solve over the Real numbers, ex...

Hello. Please look over my answers! Homework Statement a) Prove that this set is not convex: x = [1, 2] U [3, 4] c R b) Prove the intersection of two bounded sets is bounded Homework Equations for a) x = [1, 2] U [3, 4] c R The Attempt at a Solution a) A convex set is where...

Suppose $$A$$ is a set with at least two elements and $$A\times A\sim A.$$ Then $$\mathcal{P}(A)\times\mathcal{P}(A)\sim\mathcal{P}(A).$$ My attempt: I know that $$\mathcal{P}((A\times A)\cup A)\sim\mathcal{P}(A\times A)\times\mathcal{P}(A)\sim\mathcal{P}(A) \times \mathcal{P}(A).$$ How to...

In definition 2.17 of Rudin's text, he says that a set E is convex if for any two points x and y belonging to E, (1−t)x+ty belongs to E when 0<t<1. I learned that this means the point is between x and y. But I'm not able to see this intuitively. Can anyone help me "see" this?

For any sets $$A,\ B,\ C$$, $$^{(A\times B)}C\sim ^A\left(^BC\right).$$ First I could not find a formula for the required function. Then I defined the set $$D_x=\{(y,z)\in B\times C\mid f(x,y)=z\},$$ where $$f\colon A\times B\to C.$$ What do you think?

Am I right in thinking that if we have 3 sets A,B,C, then with A intersect B represented as AB, we have: A(B\C)=(AB)\C=B(A\C)?

I recently made a post on Linear and Abstract Algebra and used the following symbol $$ {\bigcup}_{\Omega \subseteq \Gamma , | \Omega | \lt \infty} $$ However, I really wanted (for neatness and clarity) to have the term $$ {\Omega \subseteq \Gamma , | \Omega | \lt \infty} $$ actually under the...

Which formula's would you use to solve each set and please show the actual formulas please thank you for the help I really appreciate it as I need to know this for my final exam tomorrow, thank you! a) solving for neither set A and neither set B b) Solving for set A or Set B, but not bothc)...

I have two questions: I have a set of data, a measured spectrum. When I model the spectrum with a function, I calculate r2=1-(\sum(y-ymodel)2/\sum(y-yavg)2). Q1) However, I have reference data now, which is what the spectrum should be. So is it right to use the same calculation on it for...

Homework Statement Determining two sets of boundary conditions for a double integral problem in the polar coordinate system. Is the below correct? Homework Equations The Attempt at a Solution There are two sets of boundary conditions that you can use to solve this problem in the polar...

Hey everyone, I have three problems that I'm working on that are review questions for my Math Final. Homework Statement First Question: Determine if R is an equivalence relation: R = {(x,y) \in Z x Z | x - y =5} and find the equivalence classes. Is Z | R a partition? Homework...

I am trying to show that an open set in [0,1] is measurable, given that [0,x] is measurable set for each x in [0,1]. So I need to show (a,b) is measurable. Using the fact that measurable sets form a sigma algebra, I have managed to show that (a,b] is measurable. So (a,b+t] is measurable for any...

Hi, I was trying to plot two sets of data on the same graph and this was my solution but this generates a graph with the x-axis in 0.5 increments where I wanted 1. How do I plot two sets of data on the same graph and customize the increments. Thanks in advance. Also, how would I make MATLAB...

Question 1) Write ⊆ or ⊄: {x/(x+1) : x∈N} ________ QNOTE: ⊆ means SUBSET ⊄ means NOT A SUBSET ∈ means ELEMENT N means Natural Numbers Q means Rational Numbers Question 2) Which of the following sets are infinite and uncountable? R - Q {n∈N: gcd(n,15) = 3} (-2,2) N*N {1,2,9,16,...} i.e...

I am currently reading Munkres' book on topology, in it he defines an open sets as follows: "If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T." Firstly, are the open sets a property of the set X or the topological...

[FONT=arial]Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

For the following example:(if possible give example or just state impossible 1) a bounded subset A of R for which sup A is not a limit point of A. An example is (0,1) union {7}. will this work? 2) a finite subset A of R that is not closed I think it is not possible. Please give some hints...

I'm having trouble working out a few details from my probability book. It says if P(An) goes to zero, then the integral of X over An goes to zero as well. My book says its because of the monotone convergence theorem, but this confuses me because I thought that has to do with Xn converging to X...

Assume that $$ f: E \to Y \,\,\, , E \subset X$$ then can we say that $$f(E^c)=f(E)^c$$ what about the inverse mapping $$f^{-1}: V \to X \,\,\, , V\subset Y$$ do we have to have some restrictions on f and its inverse ? My immediate answer is that we have to have a bijection in order to conclude...

I am reading Dummit and Foote: Section 15.2 Radicals and Affine Varieties. On page 678, Proposition 16 reads as follows: (see attachment, page 678) --------------------------------------------------------------------------------------- Proposition 16. Suppose \phi \ : \ V...

I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets. On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows: ---------------------------------------------------------------------------------------------- Definition. A map...

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment) ----------------------------------------------------------------------------------------------...

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment) ----------------------------------------------------------------------------------------------...

I am reading Dummit and Foote on affine algebraic sets and wish to create posts referring to such objects. The notation for a subset Z(S) of affine space is a "curly" Z - see attachment - bottom of page 658. Also the notion for the unique largest ideal whose locus determines a particular...

if A is a subset of B and the frontier of B is a subset of A then A=B. I am pretty sure that this is true as I drew I diagram and I think this helped. A frontier point has a sequence in the set and a sequence in the compliment that both converge to the same limit. However I'm not really...

Homework Statement Let f be a function from E to F . Prove that f is an injective function if and only if for all A and B subsets of P(E)^2. f(A\cap B)=f(A)\cap f(B) The Attempt at a Solution Well since we have "if and only if" that means we have an equivalences so for. \Rightarrow If f...

I know this is probably the most basic question imaginable so please bear with me. I did google it but I still couldn't figure it out. Say you have a function $$f(x, y, z)$$ and a point $$(x_0, y_0, z_0)$$ that satisfies the equation $$f(x, y, z) = 0$$. Does that imply that $$f(x_0, y_0, z_0)...

[FONT=arial]Definition: A subset $D$ of $\mathbb R$ is said to be discrete if for every $x\in D$ there exists $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\cap D=\{x\}$. Question: Does there exist a discrete subset $D$ of $\mathbb R$ such that the set of limit points of $D$ is an uncountable...

I would like to know if there is a general formula, and if so, what it is, for finding the $limsup$ and $liminf$ of a sequence of sets $A_n$ as $n\rightarrow \infty$. I know the following examples: **(1)** for $A_n=(0,a_n], (a_1,a_2)=(10,200)$, $a_n=1+1/n$ for $n$ odd and $a_n=5-1/n$ for $n$...

Prove that for any collection {O[SIZE="1"]α} of open subsets of ℝ, \bigcap O[SIZE="1"]α is open. I did the following for the union, but I don't see where to go with the intersection of a set. Here's what I have so far: Suppose O[SIZE="1"]α is an open set for each x \ni A. Let O=...

Homework Statement I need to answer a bunch of topological questions based on the cartesian product of two sets, but I'm not entirely sure how to graph them out. I have A = [1,2)U{3} and B = {1, (1/2), (1/3), ...}U[-2,-1). S = A x B, and I need the graph of S. Could anyone help me with...

How many elements does each of these sets have where a and b are distinct elements? (with steps please) a) P({a,b{a,b}})b)P({∅,a,{a},{{a}}}) c)P(P(∅)) *i have tried to solve them but i am a little bit confused... Thanks in advance :)

Obviously countable sets of R are Lebesgue null sets - but are Lebesgue null sets countable?

Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism. If X is a dense set in H, then is A(X) a dense set in K? Any references to texts would also be helpful.

Homework Statement Theorem: Let S be a subset of the metric space E. Then S is closed iff whenever p1,p2,p3,... is a sequence of points of S that is convergent in E, we have lim n→∞ p_n ∈ S. Homework Equations The Attempt at a Solution I am having trouble understand the "if"...

Homework Statement . Let ##f: (X,d) → (Y,d')## a uniform continuous function, and let ##A, B \subseteq X## non-empty sets such that ##d(A,B)=0##. Prove that ##d'(f(A),f(B))=0## I've been thinking this exercise but I don't have any idea where to or how to start, could someone give me a hint?

Homework Statement I am very uncomfortable with the concept of "level sets" and don't quite get what it means. Homework Equations y = 2(x1)^2 - (x1)(x2) + 2(x2)^2 y = 2x1^(1/2) * x2^(1/2) The Attempt at a Solution I'm not even sure where to start... Can anyone please give me...

Hi everyone, new member here. Anyway, for my astronomy class my professor wants us to calculate the zone time the center of the sun sets on the winter solstice, at 40 degrees north latitude, and I'm a bit stuck. I know on the winter solstice, the declination of the sun is -23.5 degrees and the...

I have a (probably trivial) question about coordinate charts. I've been studying Sean Carroll's lecture notes on General Relativity. I'm on my second re-read and I'm trying to make sure I understand the basics properly. I hope the terminology is correct - this is my first use. Carroll cites...

1. Homework Statement . Let A be a chain and B a partially ordered set. Now let f be an injective function from A to B and suppose that if a,b are elements of A and a≤b, then f(a)≤f(b). Prove that f(a)≤f(b) implies a≤b. 3. The Attempt at a Solution . I want to check if this proof by...

open

prove the followinga. prove that if $A\cap B=\emptyset$, then $(A\times C)\cap (B\times C)=\emptyset$ b. $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$ i don't have any idea how i would start proving this. can you give me some techniques on proofs.

I've been struggling for a few minutes with this basic thing and I want to make sure I got it right, given A, B being disjoint, We know that P(A and B) = 0 However, if they are independent then P(A and B) = P(A) x P(B) Then if P(A) is finite non zero and P(B) is finite non zero, how...

the associative axioms for the real numbers correspond to the following statements about sets: for any sets A, B, and C, we have $(A\cup B)\cup C=A\cup (B\cup C)$ and $(A\cap B)\cap C=A\cap (B\cap C)$. Illustrate each of these statements using Venn diagrams. can you show me how to draw the...

My basic question is this: does an arbitrary open set in ℝ2 look like a bunch of regions bounded by continuous curves, or are there open sets with weirder boundaries than that? Let me state my question more formally. A Jordan curve is a continuous closed curve in ℝ2 without self-intersections...

Written by micromass: The newest challenge was the following: This was solved by HS-Scientist. Here's his solution: This is a very beautiful construction. Here's yet another way of showing it. Definition: Let ##X## be a countable set. Let ##A,B\subseteq X##, we say that ##A## and...