Sets Definition and 1000 Threads

1. Homework Statement Use set builder notation to give a description of each of these sets. a) { 0,3,6,9,12 } b) { -3, -2, -1,0, 1, 2, 3 } c) { m,n,o,p } 3. The Attempt at a Solution X={x l x is an odd possitive multiplier of 3 less than 12 } X is supposed...

Is it true that every open set contains a compact set?

Hello, I have a problem where I have a set of positive real numbers and must partition this set into two new sets such that: 1. The sum of the values in each set is as close as possible to the sum of the values in the other set. i.e. the difference is as close to zero as is possible...

Homework Statement Show that S = [0,1) is not compact by giving an closed cover of S that has no finite subcover. Homework Equations The Attempt at a Solution I know that S is not compact because it is an open not a closed set even though it is bounded. But I am completely...

Homework Statement although I'm supposed to provide explanations for these, i just want to see if my intuition about them is correct.. 1) For any set A \subseteq R, \overline{A}^{c} is open 2) If a set A has an isolated point, it cannot be an open set 3) Set A is closed if and only if...

I'm taking real analysis and struggling a bit. In class today our professor was saying something about how a function may not be continuous on a non compact set or something, but anyway, he drew the closed interval from 0 to 1 but looped one end back to the middle of the interval. __...

Homework Statement Show if K contained in R is compact, then supK and inf K both exist and are elements of K. Homework Equations The Attempt at a Solution Ok we proved a theorem stating that if K is compact that means it is bounded and closed. So if K is bounded that means...

are there lots of examples of non-measurable sets? the one that seems to be in most textbooks involves a type of addition mod an irrational number with equivalence classes, etc etc, which in some books is done geometrically as rotations of a circle through an irrational angle. that example was...

Suppose f:[a,b]--> R and g:[a,b]-->R. Let T={x:f(x)=g(x)} Prove that T is closed. I know that a closed set is one which contains all of its accumulation points. I know that f and g must be uniformly continuous since they have compact domains, that is, closed and bounded domains. Now T is the...

Homework Statement Let E be a nonempty subset of R, and assume that E is both open and closed. Since E is nonempty there is an element a \in E. Denote the set Na(E) = {x > 0|(a-x, a+x) \subsetE} (a) Explain why Na(E) is nonempty. (b) Prove that if x \in Na(E) then [a-x, a+x] \subset...

I'm looking over some stuff from metric spaces and I came across the familiar theorem: Let \left(X,d\right) be a metric space and let \left\{ U_\alpha \right\}_{\alpha \in A} be a family of open subsets of X. Then the union of the family \left\{U_\alpha\right\}_{\alpha \in A } is an open...

Homework Statement Let S be a set of things and let P be the set of subsets of S. For A,B in P, define A*B=[(S-A)intersection B] union [A intersection (S-B)] I'm suppose to show that (p,*) is commutative, find the identity, and given that A is a subset of S, find the inverse of A. How do i...

Homework Statement Let S be a set of thing and let P be the set of subsets of S. For A,B in P, define A*B=[(S-A)intersection B] union [A intersection (S-B)] Homework Equations Consider the set S={alice, bob, carol, don, erin, frank, gary, harriot}. Using the set operation * find the...

Homework Statement Let S be a set of things and let P be the set of subsets of S. For A, B in P define A*B = ((S-A) intersect B) union (A intersect (S-B)). Need to show that (P,*) is commutative and the group identity. Homework Equations The Attempt at a Solution only...

I've been having trouble partitioning number systems into sets. The complex and rational number systems blow me away, so I'll stick with all reals, integers, and naturals for now. Homework Statement 1a) With five sets of infinitely many positive integers, partition the set of all real...

Hello, Let's say I have 500 boxes and 500 hundred non-identical items. I would like to have sets of 40, chosen among those 500 hundred items and my objective is to keep the number of same items in any 2 boxes at a minimum. 1. What would be that minimum number of common items? 2. If...

Homework Statement Prove P(A) \cup P(B) \subseteq P(A \cup B) Homework Equations The Attempt at a Solution I started out by assuming that A = \left\{a\right\} and B=\left\{b\right\}. So then P(A) \cup P(B) = \left\{\left\{a\right\},\left\{b\right\},null\right\} and P(A...

Homework Statement Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows: d(E,F) = inf_{x\in E , y\in F} | x - y | (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E...

Homework Statement Let S={p+q\sqrt{2} : p,q \in Q} and T={p+q\sqrt{3} : p,q \in Q}. Prove that S\capT = Q. Homework Equations See above. The Attempt at a Solution I was thinking possible using S\capT=Q S + T - S\cupT = Q But I have no idea how to combine them? I don't believe it's...

Fist, this is for a personal project of mine, I've been out of school for some time, and all my old math skills seem to have abandoned me. For that reason, and because I have to translate anything I learn into functions compatible with c#'s math library, it'd probably be best to keep any answers...

The question I had was to show that if a function is continuous, open and bijective then it is a homeomorphism. At first I said "no" because I thought of the example showing that [0,2pi) is not homeomorphic to the unit circle S. I knew that f(x)=(sinx,cosx) is a continuous bijection whose...

Homework Statement Theorem: If S is any bounded set in n space, and d>0 is given, then it is possible to choose a finite set of points pi in S such that every point p existing in S is within a distance d of at least one of the points p1, p2, ..., pm. Prove this theorem assuming that the...

Homework Statement An open set in the complex plane is, by definition, one which contains a disc of positive radius about each of its points. Prove that: (a) the intersection of two open sets is an open set (b) the union of arbitrarily many open sets is an open set Homework Equations...

(1/2,1/3),[1/2,1],[0,1/2] which are open in the subspace topology of the subspace [0,1] of the lower limit topology on R

Homework Statement Hi, new to the Physics Forum and desperately need some help with a math analysis problem... Prove that {x|x>1} and {x|0<x<1} are equivalent sets by writing a function and show that it is one-to-one and onto. Homework Equations The Attempt at a Solution

Homework Statement Show that any two disjoint nonempty open sets are mutually separated. Show that any two disjoint nonempty closed sets are mutually separated. Homework Equations The Attempt at a Solution Let A and B be two disjoint nonempty open sets: Assume A and B...

Homework Statement Let U_n = {all p = (x, y) with |p - (0, n)| < n}. Show that the union of all the open sets U_n, for n = 1, 2, 3, ..., is the open upper half plane. Homework Equations The Attempt at a Solution U_n describes points p whose distance from a set point on the...

1. If X is an infinite set and x is in X, show that X ~ X \ {x} A~B if there exists a one-to-one function from A onto B. Attempt at a solution I'm pretty much completely stumped on this problem. I know that since X is infinite then it contains a sequence of distinct points. So...

Homework Statement Definition:. Let A and B be sets. The Cartesian product AXB of A and B is the set of ordered pairs (a, b) (3) Assume that A and B are countable sets. Prove that the Cartesian product A x B is countable. Homework Equations The Attempt at a Solution I know...

I need to prove that the following is an open subset of R^2: \left\{(x,y)\inR^{2}|\sqrt{x^2+y^2}<1} I think the substition r=min{sqrt[x^2+y^2],1-sqrt[x^2+y^2]} works, but I'm stuck on how to take it from that to showing that the distance between X0 and X1 is less that r, and more...

Homework Statement Show that the union of convex sets does not have to be convex. Homework Equations The Attempt at a Solution Is it enough to just show a counterexample? Or is that not considered a complete proof? My example is...S = {1} and T = {2}.

Is it an oversimplification to say a countably infinite set is an algebra while an uncountably infinite set is a topology?

I've two problems: Given are the two sets A = \left \lbrace (x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4} \mid x_{0}^{2} = \vec{x} \, ^{2}, x_{0} \geq 0 \right \rbrace and B = \left \lbrace (x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4} \mid (k_{0} - x_{0})^{2} = (\vec{k} -...

Russels paradox says that the set of all sets not contained in themselves, i.e. x=\{ y \ : \ y \not \in y \} neither is or is not contained in itself. The set can be created based on an axiom saying that "The set of objects with a property Q exists". Let`s assume Q : "Is not contained within...

I read in a book on groups and representations that any transitive G-set is isomorphic to the coset space of some subgroup of G. Does this mean we can determine all transitive G-sets up to isomorphism simply by finding all subgroups of G? Just want to make sure that if this is the case that I...

Proposition: Let \{A_n\}_{n\in I} be a family of countable sets. Prove that \bigotimes_{i=1}^n A_i is a countable set. Proof: Since \{A_n\}_{n\in I} are countable, there are 1-1 functions f_n:A_n->J (J, the set of positive integers) Now let us define a function...

[b]1. I am having trouble grasping cauchys theorm. To my understanding the integral for any closed contour in which f(z) is path connected and analytic =0. Homework Equations Example 1: Now the integral around the unit circle for 1/z. In the unit circle 1/z is anlaytic for all z, but...

Homework Statement Prove that for every set S, S \subseteq S. Use 'proof by cases'. Homework Equations A \subseteq B iff {X: X \in A --> X \in B} The Attempt at a Solution I know that A is a subset of B if every element of A is also an element of B. In the case of S \subseteq...

1. The problem statement Need to prove that the set of bijections from N to N is uncountable.2. The attempt at a solution I'm not really sure how to proceed here but what I did so far is this... f(2i) = { 2i+1, if fi(2i)=2i 2i , if fi(2i)not equal to =2i }Not very sure what...

I'm talking about E \times F, where E,F \subseteq \mathbb{R}^d. If you know E and F are compact, you know they're both closed and bounded. But how do you define "boundedness" - or "closed", for that matter - for a Cartesian product of subsets of Euclidean d-space? The only idea I've had is...

Homework Statement There is only a small issue that i am confused about... If we have a set \left(-\frac{1}{n},\frac{1}{n}\right), where n is a natural number. If we want to find the intersection of all such sets, my question is whether the result will be the set containing only...

Hi, I would like to know if you guys know a good book on sets groups and relations, preferably with lots of exercises. I believe that I am on a beginner level, but I already know all basic concepts, so the text is not that important. It would be even better if it is available online hehe! Thks!

Given a set of sets such that A_{i}\subset{C}. Every subset has a countable infinity of elements. I want to create a set W such that it contains exactly one element from each subset A_{i}. I presume I can do this by describing the intersect of W with every subset A_{i} as containing exactly one...

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Hi all, Does anybody know of any graduate level math texts with extremely difficult problem sets (olympiad level or harder)?

Hi! I'd like to ask the following question. Does it make sense to take unions and intersections over an empty set? For instance I came across a definition of a topological space which uses just two axioms: A topology on a set X is a subset T of the power set of X, which satisfies: 1...

Homework Statement Let A,B,C be sets. (a) Show that: A \cup B = (A\B)\cup(B\A)\cup(A \cap B) (b) Show that: A \times (B\C) = (A \times B) \ (A \times C) Homework Equations The Attempt at a Solution For part (a) I need to prove the definition of a union. I think in...

Homework Statement Let X by a completely regular space and let A and B be closed, disjoint subsets of X. Prove that if A is compact, then there is a continuous function f : X --> [0,1] such that f(A) = {0} and f(B) = {1}. The attempt at a solution Let {U} be an open covering of A, U_1...

Two sets are equal iff they contain the same elements. I would argue that two sets that have the same elements are identical as well as equal and that there is a difference between identity and equality. In general {2,3}={3,2} if neither set is defined to be ordered. However obviously {5} \neq...

I read this: A class T of subsets of X is called a topology on X if it satisfies these 2 conditions: 1. The union of every class of sets in T is a set in T. 2. the intersection of every finite class of sets in T is a set in T. The sets in the class T are called the open sets of the...