Sets Definition and 1000 Threads

Given a totally finite measure μ defined on a \sigma-field X, define the (pseudo)metric d(A,B)=μ(A-B)+μ(B-A), (the symmetric difference metric), it can be shown this is a valid pseudo-metric and therefore the metric space (X',d) is well defined if equivalent classes of sets [A_\alpha] where...

let A={z|z^6=√3 + i} B=(z|Im(z)>0} and C={z|Re(z)>0} find A∩B∩C the part previous to this qn asks me to find the roots of z^6 and I've already down that. but i have no idea how to proceed with this, so do i draw my unit ciorcle with the hexagon and then follow to see what regions satisfies with...

Find two open sets A and B, such that A is subset of B, A is not equal to B, and m(A)=m(B) Can I use these two sets? A=(0,2) B=(0,1) U (1,2) thanks

Could someone please show that an open ball is open where the definition of "open" is: A set is open if for each x in U there is an open rectangle A such that x in A is contained in U. Where an open rectangle is (a_1,b_1)×…×(a_n,b_n). I also realize that one can use rectangles or balls, but I...

I barely started out learning on my own about proofs from this book called A transition to advanced math 2nd edition by chartrand. I am having trouble understanding what an indexed set is and the notation. Is there any online resources I can use to help me understand this concept?

Homework Statement Decide whether the following propositions are true or false. If the claim is valid supply a short proof, and if the claim is false provide a counterexample. a) An arbitrary intersection of compact sets is compact. b)A countable set is always compact. The Attempt at a...

Homework Statement Let \sigma (E)=\{(x,y):x-y\in E\} for any E\subseteq\mathbb{R}. If E has measure zero, then \sigma (E) has measure zero. The Attempt at a Solution I'm trying to show that if \sigma (E) is not of measure zero, then there exists a point in E such that \sigma (\{e\}) that has...

The stupid question of the day. If S is the real interval (-infinite, 5], and I can find a metric d so that (S,d) is a metric space, then, is, for example, (4, 5] an open set in (S,d) ? I say this because, the way I'm reading the definition of an open ball, the open ball B(5,1) is the...

Homework Statement If A is a set that contains a finite number of elements, we say A is a finite set. If A is a finite set, we write |A| to denote the number of elements in the set A. We also write |B| < ∞ to indicate that B is a finite set. Denote the sets X and Y by X = {T : T is a proper...

Wiki says: Isn't this exactly what every A/D converter does? For a graph of Vin to digital output it basically approximates the nearest digital value to the continuous signal -> So I don't see the difference between them.

When you spend hours and days tediously plugging away at the mathematics of a problem, you lose sight of the actual physics of the problem (in addition to losing sight of what you found interesting about physics in the first place). The problem statements are always innocuous, but as soon as...

How would one show that if there is a number c for which g'(c)=0, then every point on the level set {(x,y)|H(x,y)=c} is a degenerate critical point of f? I know that the question may seem vague, but this is the question as it was given to me by my professor. It is something to think about...

"Adding" 2 open sets Homework Statement I'm trying to prove that If both S and T are open sets then S+T is open set as well.Homework Equations S+T=\{s+t \| s \in S, t \in T\}The Attempt at a Solution S+T is open if every point x_0 \in S+T is inner point. Let x_0 be a point in S+T, so there...

Homework Statement Show if a set is infinite, then it can be put in a 1-1 correspondence with one of its proper subsets. Homework Equations This was included with the problem, but I am sure most already know this. A is a proper subset of B if A is a subset of B and A≠B The...

The original problem is as follows: IF E,F are measurable subset of R and m(E),m(F)>0 then the set E+F contains interval. After several hours of thought, I finally arrived at conclusion that If I can show that m((E+c) \bigcap F) is nonzero for some c in R, then done. But such a...

I tried very long time to show that For closed subset A,B of R^d, A+B is measurable. A little bit of hint says that it's better to show that A+B is F-simga set... It seems also difficult for me as well... Could you give some ideas for problems?

Folks, I am looking at my notes. Wondering where the highlighted comes from. Prove that a finite orthogonal set is lineaarly independent let u=(x_1,x_2,x_n) bee an orthogonal set set of vectors in an ips. To show u is linearly independent suppose Ʃ ##\alpha_i x_i=0## for i=1 to n...

Homework Statement Let W be a subspace of R^n and let A be a subset of R^n. Then A spans W if and only if <A> = W (<A> is the set of all vectors in R^n that are dependent on A). Prove it. Homework Equations The Attempt at a Solution Ok the book goes likes this; First, it proves...

I have some questions about spanning sets 1. Why does empty set spans the zero subspace? 2. Why is this true: Since any vector u in A is dependent on A, A⊆<A>? (<A> is the set of all vecotrs in R^n that are dependent on A)

Homework Statement Show the following sets are countable; i) A finite union of countable sets. ii) A countable union of countable sets. Homework Equations A set X, is countable if there exists a bijection f: X → Z The Attempt at a Solution Part i) Well I suppose you could start by considering...

Hello. I was wondering whether a non-empty dense set has any isolation points. From my understanding, when a set is dense you can always find a third point between two points that is arbitrarily close to them so any ball you "create" around a point will contain another point hence a non-empty...

Proving the "triangle inequality" property of the distance between sets Here's the problem and how far I've gotten on it: If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference. And D(A, B) = m^*(S(A, B)), which is the outer measure of...

Homework Statement Give an example of an infinite collection of nested open sets. o_1 \supseteq o_2 \supseteq o_3 \supseteq o_4 ... Whose intersection \bigcap_{n=1}^{ \infty} O_n is closed and non empty. Homework Equations A set O \subseteq \mathbb{R} is open if for all points...

Homework Statement The question is "Prove f(\bigcap_{\alpha \in \Omega} A_\alpha) \subseteq \bigcap_{\alpha \in \Omega} f(A_\alpha) where f:X \rightarrow Y and \{A_\alpha : \alpha \in \Omega\} is a collection of subsets of X. Also, prove the statement's equality when f is an injective...

Hi all! I am searching for an algorithm (most likely already present in the literature) that could solve the following problem: Instance: Properties of sets of elements and relations between sets of elements Question: Find the closure of the properties and relations Possible properties...

Now I'm not a PMI (perpetual machine inventor). In fact I'm quite convinced that there is no such thing as that. But a while ago, I saw the schematics of a perpetual machine that is hard to debate. Well this is how the machine worked. The inventor argued that if you have two magnets as...

Homework Statement Are the following regions in the plane (1) open (2) connected and (3) domains? a. the real numbers; b. the first quadrant including its boundary; c. the first quadrant excluding its boundary; d. the complement of the unit circle; f. C \ Z = {z ∈ C : z \notinZ}...

Hi, When you have to calculate the rising or setting time of a celestial body, you have to handle with hour angle and sidereal time. Sidereal time for the rising is given by T = alpha - H and by T = alpha + H for the setting (alpha = right ascension). Why - H in one hand, and + H on the...

This is not a homework problem, just something I was thinking about. In a general metric space, is it true that every closed set can be expressed as the intersection of an infinite collection of open sets? I don't really know where to begin. Since the finite intersection of open sets is open...

Homework Statement This is not really coursework. Instead, this is some sort of curiosity and proposition formulation rush. Then the initial questions are that if this is a valid result that is worth to be proven. Let X,Y be metric spaces and X\times Y with another metric the product metric...

whats an infinite intersection of open sets? how is it different from finite intersection of open sets and why is it a closed set in the case of ∞ intersection but open in case of finite. To quote kingwinner, is it being defined as a limit? it really does look look like a limit in the case...

ℝHomework Statement I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated: (i) Prove f: V→W is affine (where V and W are real vector space)...

Homework Statement Find (X,d) a metric space, and a countable collection of open sets U\subsetX for i \in Z^{+} for which \bigcap^{∞}_{i=1} U_i is not open Homework Equations A set is U subset of X is closed w.r.t X if its complement X\U ={ x\inX, x\notinU} The Attempt at a Solution Well...

Homework Statement Show by induction that if the finite sets A and B have m and n elements, respectively, then (i) A X B has mn elements; (ii) A has 2m subsets; (iii) If further A \cap B = \varphi, then A \cup B has m+ n elements. NOTE : I am only interested in the (iii) section of...

I'm a little bewildered when reading about these Julia sets. From the definition a Julia set is the closure of all repelling periodic points of a complex map f. However I read that a Julia set always contain periodic and non-periodic points. Wasn't the definition including only periodic points...

Homework Statement Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s \in S and t \in T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T. The Attempt...

Homework Statement I have these two sets: Pairwise, (1, 1) (2, 4) (3, 9) (4, 16). Clearly this is just squared. How can I get a function relation with like: (1, 1) (2, 3) (3, 9) (4, 10) or like (1, 1) (2, 5) (3, 12) (4, 22) Homework Equations The Attempt at a...

Suppose I define sets D_n = \lbrace x \in [0,1] | x has an n-digit long binary expansion \rbrace . Now consider \bigcup_{n \in \mathbb{N}} D_n. This is just the set of Dyadic rationals and therefore countable for sure. Now for the question: is this equal to \bigcup_{n = 0}^{\infty} D_n...

Homework Statement Prove that every infinite subset contains a countably infinite subset. Homework Equations The Attempt at a Solution Right now, I'm working on a proof by cases. Let S be an infinite subset. Case 1: If S is countably infinite, because the set S is a subset...

Homework Statement If s (0,1), find |P(S)|, |P(P(S))|, |P(P(P(S)))| Homework Equations The Attempt at a Solution |P(S)| = {(0), (1), (0,1), ∅} = 4 |P(P(S))| = {...} = 16. |P (P(P(S)))| = {...} = 16 ^4 ...but how? as my lecturer explained it, it come from pascals...

What is the probability that a number selected from 0-9 will be the same number as one randomly selected from 0-4? Relevant equations: $$P(A \cap B) = P(A)*P(B|A)$$ I used the equation above, using A as the event that the number selected from 0-9 will be between 0 and 4, and B as the event that...

For a subset which is both closed and open (clopen) does its closure equal its interior?

Homework Statement Suppose A and B are both countably infinite sets. Prove there is a 1-1 correspondence between A and B. Homework Equations The Attempt at a Solution Since A is countably infinite, there exists a mapping f such that f maps ℕto A that is 1-1 and onto...

Homework Statement Suppose that A and B are both countably infinite sets. Prove that there is a one to one correspondence between A and B. Homework Equations The Attempt at a Solution By definition of countably infinite, there is a one to one correspondence between Z+ and A and...

Homework Statement Let Ts denote the set of points in the x; y plane lying on the square whose vertices are (-s; s), (s; s), (s;-s), (-s;-s), but not interior to the square. For example, T1 consists of the vertices (-1; 1), (1; 1), (1;-1), (-1;-1) and the four line segments joining them...

Homework Statement prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R Homework Equations The Attempt at a Solution is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between...

This research project is to help me (I'm an undergraduate) get my head around this topic. It is concerned with affine subsets of a vector space and the mappings between them. As an application, the construction of certain fractal sets in the plane is considered. It would be considered pretty...

Hello all, I am having trouble with a homework problem. The problem is as such: Let a = {zn = (xn,yn) be a subset of ℝ2 and zn be a sequence in ℝ2 such that xn ≠ xm and yn ≠ ym for n≠m. Let Ax and Ay be the projections onto the x and y-axis (i.e. Ax = {xn} and Ay = {yn}. Assume that the...

My proof class just took a turn for the worst for me - I don't understand this. First, the notation is extremely confusing to me, I need help to make sure I'm getting this. If An is some set for some natural number n such as [-n, n]. Then (script A) the collection is the set of all An...

A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Does not this imply that every open set is compact. Because let F is open, then F= F \bigcup ∅. Since F and ∅ are open , we obtained a finite subcover of F. Am I missing something here?