Sets Definition and 1000 Threads

Homework Statement Okay, so we've moved on from talking about R^n to talking about general metric spaces and the differences between the two. We're given that X (a metric space) satisfies the Bolzano-Weierstrass Property and that A and B are disjoint, compact subsets of X. Dist(A,B) is defined...

Linear Algebra- Basis Homework Statement Is {e1,e2} a basis for R3 ? Homework Equations The Attempt at a Solution I know that {e1,e2,e3} is a basis for R3 same here, Is this one a basis for R3 {(1,1,2)T,(2,2,5)T} I know that {(1,1,2)T,(2,2,5)T,(3,4,1)T} is a basis...

Homework Statement Q1) Assuming that |R|=|[0,1]| is true, how can we rigorously prove that |R2|=|[0,1] x [0,1]|? How to define the bijection? [Q1 is solved, please see Q2] Q2) Prove that |[0,1] x [0,1]| ≤ |[0,1]| Proof: Represent points in [0,1] x [0,1] as infinite decimals...

Hi everyone. I wasn't really sure where to put this thread so I stuck it here, which seems the closest fit. Anyway I've been thinking about this for too long: what characterises the unit ball of a norm? Let's be specific: consider a finite-dimensional vector space, which may as well be...

Let K1, K2, K3, . . . be an infnite sequence of sets, where each set Kn is countable. Prove that the union of all of these sets K = Union from n=1 to infinity, Kn is countable. I tried to start, but I don't even understand the question Need some idea on how to start

Homework Statement So I'm trying to find a basis for the space that is spanned by the given vectors. {(1,0,0,1) (-2,1,-1,1) (6,-1,2,-1) (5,-3,3,-4) (0,3,-1,1)} These are written as column vectors. Homework Equations None really (that I know of) The Attempt at a Solution So I...

Hi, I have a few questions because I'm watching a lecture on real analysis & I'm a little bit unsure of a few things. I have them in point form for your convenience in answering. http://www.youtube.com/watch?v=lMHR6d0leKA&NR=1 1. (from 2.30 in the video - no need to watch) A & B are sets &...

If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete? What if it is known that A is closed, can it then be said that B is also closed?

Hi all, I am looking for examples of infinite pairwise coprime sets. A set P of integers is pairwise coprime iff, for every p and q in P with p ≠ q, we have gcd(p, q) = 1. Here gcd denotes the greatest common divisor.(Wikipedia) Also from wikipedia the following are some examples of...

Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is (-infinity, n) or [n, infinity) it is not going to be unbounded. The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply...

A={x ∈ ℤ | x ≡ 7 (mod 8)} B={x ∈ ℤ | x ≡ 3 (mod 4)} Is A ⊆ B? Yes Since x ∈ A, then xa = 7 + 8a = 8a + 7 = 2(4a + 3) +1. And since the ∈ B are of the form xb = 3 + 4b = 4b + 3 = 2(2b + 1) + 1, both ∈ A,B are odd. A ⊆ B since the ∈ of both sets are of 2p + 1. Q.E.D. Is this correct?

Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F. Claim 1: F is contained in the clousre of F...

Homework Statement Find three disjoint open sets in the real line that have the same nonempty boundary. Homework Equations Connectedness on open intervals of \mathbb{R}. The Attempt at a Solution If this is it all possible, the closest thing I could come up with to 3 disjoint...

Hi PF, This semester I'm finally starting to enjoy problem sets. I was wondering, how similar are the skills used to solve problem sets to the skills theoretical physicists use to solve actual problems? Thanks, DoD

Homework Statement Let (X,d) be a metric space and E is a subset of X. Prove that (c means complement, E bar means the closure of E) Homework Equations N/A The Attempt at a Solution Let (X,d) be a metric space and B(r,x) is the open ball of radius r about x. Definition: Let F be...

I have this question but I don't get it at all. Here goes: Let X be {x, y, z} P(X) is the power set. For all Y,Z is an element of P(X), Y R Z where The number of elements in Y intersect Z is 1. So I worked out P(X) to be: {(null), (x), (y), (z), (x,y), (x,z), (y,z)} Then I don't...

can anyone intuitively explain me what does a borel field and a borel set mean?Why do we need a Borel field to define all our definitions in probability?

1. Define numerical equivalence of sets 2. I'm not sure how in depth the definition needs to be, how is my current def? 3. X is numerically equivalent to Y if \existsF:X\rightarrowY that is bijective or there are two injective functions f:X\rightarrowY and g:Y\rightarrowX

If A is a 3x3 Matrix and {v1, v2, v3} is a linearly dependent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly dependent set? Is this true? Can someone please explain why or why not?? What I think: I think it is true because I read that a linear transformation preserves the...

I just have a very quick (and simple) question: When trying to prove equalities like A \cup (B \cup C) = (A \cup B) \cup C, is it sufficient to note that both sets consist of all elements x such that x \in A, x \in B or x \in C? Or do I need to go through proving that each set is a subset of the...

Can someone provide me an example of three sets of integers A, B and C such that A\cupB=A\cupC, but B≠C. And also, A\capB=A\capC, but B≠C. Thanks a lot :)

1) Fact: Let X be a metric space. Then the set X is open in X. Also, the empty set is open in X. Why?? 2) Let E={(x,y): x>0 and 0<y<1/x}. By writing E as a intersection of sets, and using the following theorem, prove that E is open. Theorem: Let X,Y be metric spaces. If f:X->Y is...

Hi, I'd like to know if using small letters to represent sets violates rules? From what I've been taught capital letters are pretty much used to denote sets. Is this a strict rule?

Homework Statement List all the functions from {1,2,3} to {1,2} representing each function as an arrow diagram. Which of these functions are (a) injective, (b) surjective, (c) bijective? For each surjective function write down a right inverse. Homework Equations The Attempt at a...

I have a quick regarding a definition for linear dependence that my professor gave in class... A set of vectors {v_{1},v_{2},...v_{k}}, are considered linearly dependent provided there are scalars c_{1},c_{2},...c_{k} that are not all zero, such that c_{1}v_{1} + c_{2}v_{2} + ... c_{k}v_{k} =...

If A \subseteq B does that mean A = B which means B = A because if A is a proper \subset of B then A does not equal B right. I am wrong right?

This is not a homework problem, just a question from a discussion with my classmates about the Cantor set. The original goal is to prove Cantor set is closed. My earlier attempt is to show the complement of the Cantor set is open. Since when construct the Cantor set each time the sets removed...

hi guys... this is my first thread in this forum.. i hope i'll learn math more easily with this forum... mmm.. i've a some math homework that i must submit it this Thursday... :-p but, i don't know how to answer it... the questions is... 1. Prove that X ⊆ X U Y for all sets...

I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears. So to get to the meat of the question, given that \tau is a topology on the set X giving (\tau,X), the members of \tau are called open sets. Up to that point...

Homework Statement Let A, B, C be any sets. Prove that if C\subseteq A, then (A\cap B)\cup C = A\cap (B\cup C) Homework Equations ? The Attempt at a Solution Don't even know where to begin, If someone could point me in the right direction, that would be the best.

Homework Statement This is the Theorem as stated in the book: Let S be a subset of a metric space E. Then S is closed if and only if, whenever p1, p2, p3,... is a sequence of points of S that is convergent in E, we have: lim(n->inf)pn is in S. Homework Equations From "introduction to...

In general, if S is a connected set, can I conclude that S must be path connected? Definition 1: S is connected if it is not disconnected. A set S is disconnected if it can be written as the union of two mutually separated sets, where mutually separated sets are two nonempty sets that do not...

Let a and b be sets. Show that the following constructions are sets stating clearly which axioms you need (a) a\b. (b) A function f:a→ b. (c) The image of f. (d) Given that a and b have ranks α and β respectively, what are the maximum possible ranks of a\b, f:a→ b and the...

Homework Statement (1)If C is a set with c elements , how many elements are in the power set of C ? explain your answer. (2)If A has a elements and B has b elements , how many elements are in A x B ? explain your answer. Homework Equations The Attempt at a Solution (1)The...

Hi, I have 2 sets of data, one is 472 data points long, the other 370. They are both a function of the same variable "x", they both have the same value for "x" as the last point and the same value for "x" as the first point. I'm being asked to average the two sets of data, but obviously I...

Take the set of positive integers plus 0 and the subset of all positive integers ending in, say 5. I see no reason why there can't be a one to one correspondence between the subset and the set. Am I wrong? (I have been challenged on this assertion.) EDIT: The challenge was: The subset is a...

Let's say we have a set S, and a function f : S -> S. Now let S be endowed with a binary operation, forming a group G. Is it correct to write f : G - > G? Up to now I have been operating on the assumption that yes, although G is not technically a set, there is little harm in being sloppy and...

I need someone to help explain to me in simple terms how Julia sets work. I understand how the equations governing the Mandelbrot set work, but am finding Julia sets to be a little more complex and difficult to understand.

Homework Statement Can someone check this for me? Problem: determine which, if any of the sets if open? closed? compact? R=reals; Q=rationals and Z=integers. A= [0,1) U (1,2) is NEITHER B=Z is CLOSED C=(.5,1) U (.25,.5) U (.125, 25) U... is OPEN D={r*sqrt(2) such that r is an element of...

Let V be a linear space and u, v, w \in V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}

Hi, I was wondering if two sets are disjoint countably infinite sets why is their Cartesian product also countably infinite? Thanks!

Does anyone know of a book on Category Theory that purposely attempts to teach category theory without explicitly basing it upon set theory?

I just started learning some basic measure theory. Could someone explain the difference between \overline{F(A \times A)} and \overline{F(A) \times F(A)} where A is a finite set. Also, how would this be different in A was an countably infinite set? Thanks!

I just started learning some basic measure theory. Could someone explain the difference between \overline{F(A \times A)} and \overline{F(A) \times F(A)} where A is a finite set. Also, how would this be different in A was an countably infinite set? Thanks!

Good day all I have two lists of data A & B. The max & min for A is different from B. I want to draw circles, by specific software, representing each one of the two list of data. The max of both A&B should have the large circle and the min of both of them should have the smallest...

Homework Statement Prove that a closed rectangle A \subset \mathbb{R}^n is a closed set. Homework Equations N/A The Attempt at a Solution Let A = [a_1,b_1] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n, then A is closed if and only if its complement, \mathbb{R}^n - A, is...

Homework Statement A is a sequentially compact set, and B is a point ⃗v in Rn−A. define the distance between A and B as dist( ⃗u0 , ⃗v), where you showed that ⃗u0∈ A exists such that dist( ⃗u0 , ⃗v)≤dist(⃗u, ⃗v) for all ⃗u in A. a) Use this example to state a definition of the distance...

Homework Statement Without using the Axiom of Choice, show that if A is a well-ordered set and f : A -> B is a surjection to any set B then there exists an injection B -> A. Homework Equations The Attempt at a Solution I was wondering if the existence of the surjection from a well...

I asked this in the mathematics section, but they told me it was probably best to ask it in a different PF section. So I'm interested in getting a hold of peer-review journal articles' "original data sets". I'm curious because I want to play around with data for fun on some statistics software...

Homework Statement Let X be set donoted by the discrete metrics d(x; y) =(0 if x = y; 1 if x not equal y: (a) Show that any sub set Y of X is open in X (b) Show that any sub set Y of X is closed in y Homework Equations In a topological space, a set is closed if and only if it...