Union Definition and 218 Threads

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...

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...

Homework Statement Let A and B be non-empty bounded sets of real numbers. Show the infimum of A union B is equal to the min{infA,infB} Homework Equations If a set is bounded below, a set called S for example, there exists a number N such that x≥N for all x in S And if S is bounded...

Homework Statement If A_1,A_2...A_n are countable sets. Then the union A_1 \cup A_2\cup ...\cup A_n is countable. The Attempt at a Solution Since we know there are an infinite amount primes I will assign each element in A_1 to the first prime. I will take every element in A_1 and...

Homework Statement suppose that a metric space A is a union A = B U C of two subsets of finite diameter. Prove A has finite diameter. Homework Equations The Diameter of a metric space M is sup D(a,b) for all a,b in M. The Attempt at a Solution Really, no idea where to begin. I just...

Homework Statement A_{1}, A_{2}, A_{3},... are countable sets indexed by positive integers. I'm looking to prove that the disjoint union of these sets is countable. Homework Equations The Attempt at a Solution I can't figure out how to enter the form of the disjoint union in...

Homework Statement Let A = {x\in R | |x| >1}, B = {x\in R | -2<x<3}. Find A \cup B and A\cap B The Attempt at a SolutionI thought I might attempt this via a number line. Since I don't know how to make a number line in Latex, I'll describe it. I have A as being all of R except for the region...

I have been trying to determine the fundamental group of S^2 union a line connecting the north and south pole by using the Seifert Van Kampen Theorem. But every time I try and pick my two subsets U and V they are either not open or not arcwise connected or their intersection isn't particularly...

So I have to find an infinite union of closed sets that isn't closed. I've thought of something that might work: \bigcup[0,x] where 0\leq x<1. Then, \bigcup[0,x] = [0,1), right?

I have to prove that the arbitrary union of open sets (in R) is open. So this is what I have so far: Let \{A_{i\in I}\} be a collection of open sets in \mathbb{R}. I want to show that \bigcup_{i\in I}A_{i} is also open... Any ideas from here?

I am struggling with combining infinite unions with infinite intersections, the problem i have is to show that, for Sets Aij where i,j \inN (N=Natural Numbers) ∞...∞ \bigcup ( \bigcap Aij) i=0 j=0 is equal to ...∞ \bigcap{(\bigcupAih(i):h\inNN} ...

Homework Statement Prove: If E1, · · · , Ek are the disjoint equivalence classes determined by an equivalence relation R over a set X, then (a) X = union of disjoint equivalence classes Ej (b) R = union of disjoint (Ej x Ej) Homework Equations R is a subset of X x X The Attempt at a Solution...

Homework Statement I need to expand this: P(A∪B∪C∪D) A,B,C,D are not disjoint. Homework Equations The Attempt at a Solution P(A∪B∪C∪D) = P(A) + P(B) + P(C) + P(D) - P(A∩B) - P(A∩C) - P(B∩D) - P(C∩D) + P(A∩B∩C∩D) Is that right

I'm trying to read this book "Automata, Computability, and Complexity" by Elaine Rich and on page 75 it defines this function: \delta'(Q,c) = \cup\{eps(p):\exists q\in Q((q,c,p)\in\Delta)\} I've never seen the union operator used in this way. What does it mean? Apologies if this is in the...

Prove that a vector space cannot be the union of two proper subspaces. Let V be a vector space over a field F where U and W are proper subspaces. I am not sure where to start with this proof.

Hi, All: I am trying to understand the formal machinery leading to a proof that the homology of the disjoint union of spaces is the disjoint (group) sum of the homologies of the respective spaces; the idea seems clear: if a cycle bounds in a given space Xi, then it will bound in the...

Homework Statement If A, B, and C are subsets of the set S, show that A^C \cup B^C = \left(A \cap B\right)^C Homework Equations A^C = \{x \in S: x \not \in A\} A\cup B = \{x \in S:\; x \in A\; or\; x\in B\} A\cap B = \{x \in S:\; x \in A\; and\; x\in B\} The Attempt at a Solution...

Let X be a non-empty set, and let S contain all countable subsets of X. Partially order S by inclusion. Let C be a totally ordered subset ("chain") of S, and let U = \cup_{E \in C} E It appears that U is not always countable: if it were, U would be an upper bound of the chain C, and U would...

Hi, I am stuck with the following proofs. In metric space here, A,B,C are subset of metric space (X,d) and C is bounded Problem 1.) d(A,B) <=d(A,C)+d(B,C)+diam(C) Problem 2.)|d(b,A)-d(c,A)| <= d(b,c) where 'b' belongs to 'B' and 'c' belongs to 'C'. Problem 3)- diam(A U B)<= diam A+...

Hello, I know one proof of this well known theorem that assumes on the metric of R being the standard metric. Does this result generalize to arbitrary metrics on R? thank you

Hi everyone, I came across a problem that requires knowing this fact. But can any open set in R^n be expressed as the countable union of "cubes". That is subsets of the form (a_1,b_1) \times ... \times (a_n, b_n) .

There the well known theorem that every open set (I'm talking about R here with standard topology) is the union of disjoint open intervals. Now, looking at the geometry, it seems that between any two adjacent open intervals which are in the union constituting our open set there is a closed...

Consider the regular gramma G1 (seen below as S1) and the grammar G2 (seen below as S2). Give a left-linear grammar of L(G1) U L(G2) S1->abA A->baB B->aA | bb S2->AS2 | λ A->aaB B->bB | ab I know that S1 is a regular right-linear grammar which can be changed into a left-linear...

If U, U ′ are subspaces of V , then the union U ∪ U ′ is almost never a subspace (unless one happens to be contained in the other). Prove that, if W is a subspace, and U ∪ U ′ ⊂ W , then U + U ′ ⊂ W . This seems fairly simple, but I am stuck on how to go about proving it.

Homework Statement Let U and V both have the same cardinality as R (the real numbers). Show that U\cupV also has the same cardinality as R. Homework Equations The Attempt at a Solution Because U and V both have the same cardinality as R, I that that this means \exists f: R\rightarrowU that is...

A Union of a FINITE collection of closed sets is closed. But if it is an infinite collection? Can someone provide an example please?

If you take the intersection of non empty open subsets in Rn as n tends to infinity, such that U_1 \supseteq U_2 \supseteq U_3... Is it empty?

Question: Given that any open subset E of the set of real numbers is a disjoint union of open intervals. Is E a countable union of disj. opn intervls. Answer: Yes it is. to show this we need to find a Bijection from the set of natural numbers to E. E = disjoint U_(i in N) of (a_j ...

union proof due at midnight! (∃y)(∀x)(x ∈ y) ↔ (x ∈ a ∨ x ∈ b)) How do you prove this??

In my discrete math book there is half a page with very formal explanation of the big Union notation and two very short examples without guidance so I have a hard time understanding what goes on. Here's a http://img525.imageshack.us/img525/8507/unionl.jpg" . I know the Summation formula and I...

I know that a countable union of countable sets is countable, and that a finite product of countable sets is countable, but even a countably infinite product of countable sets may not be countable. Let X be a countable set. Then X^{n} is countable for each n \in N. Now it should also be true...

When I say A disjoin union B, does it have a truth value, like A subset B, or is it a set construction notation, like union?

I am trying to calculate the homology groups of the Klein bottle. I want to use the Mayer-Vietoris sequence with the Klein bottle decomposed as the union of two Mobius bands (A and B which are homotopic equivalent to circles), now AUB is the Klein bottle, but I don't understand how according to...

Homework Statement Prove if S1 and S2 are both open then S1 \capS2 is also open Homework Equations S1 is open means boundary(S1) \subset S1c Same for S2 pThe Attempt at a Solution We want to prove boundary(S1\capS2) \subset (S1 (intersection) S2)c Then idunno how to...

Homework Statement As the title suggests, Let {Aj} be a collection of path connected subspaces of some space X, and let the intersection of these subspaces be nonempty. Is U Aj path connected? The Attempt at a Solution Again, my answer would be no, in general. But, since their...

I hope that someone can help me with the following problem: Problem: Proof by induction that: A1 \cup A2 \cup...\cupAn=(A1-A2)\cup(A2-A3)\cup...\cup(An-1-An)\cup(An-A1)\cup (A1\capA2\cap...\capAn)

Homework Statement Show that if E \subseteq R is open, then E can be written as an at most countable union of disjoint intervals, i.e., E=\bigcup_n(a_n,b_n). (It's possible that a_n=-\inf or b_n=+\inf for some n.) Hint: One way to do this is to put open intervals around each rational point...

Let E be a subset of R2 that is non-empty, compact, and connected. Suppose furthermore that E is the union of a countably infinite number of almost disjoint closed cubes {Ri} with non-zero volume. Is there anything interesting about this set, particularly its boundary? Can it have infinite...

Homework Statement Suppose f is a function with sets A and B. 1. Show that: I_{f} \left(A \cap B\right) = I_{f} \left(A\right) \cap I_{f} \left(B\right) Inverse Image of F (A intersects B) = Inverse Image of F (A) intersects Inverse Image of B. 2. Show by giving a counter example that...

Help! Union Labor at Convention Center I was hoping someone could help me figure out the following: If I have a 40 lb trough filled with 250- 12oz sodas and 40 lbs of ice on top of a sico table on wheels that weighs 125lbs...how can i figure out if a union employee can push this to our events...

Homework Statement Show that if A_{1}, A_{2},... are countable sets, so is A_{1}\cup A_{2}\cup... Homework Equations The Attempt at a Solution Part one of the question is okay, I would like to believe I can handle that but, part B, I am not so sure. My solution is as follows ( using the...

Homework Statement I am trying to solve a question from Abstract Algebra by Hernstein. Can anyone give me hint regarding the following: Show that a group can not be written as union of 2 (proper) subgroups although it is possible to express it as union of 3 subgroups? Thanks...

Hello all, I have the following question regarding the interchange between union and intersection. \cup_{q < t} \cap_{s > q} A_{s} = \cap_{s<t} \cup_{q<s} A_{q} = \cup_{q < t} A_{q} Am I correct? Also, can anyone provide me some more resources regarding this kind of interchange in...

Homework Statement 1.Given a set T we say that T serves as an index set for family F={Aa} of sets if for every a in T there exists a set Aa in family F. 2. By the union of the sets Aa, where a is in T, we mean the set {x l x\inAa for at least one a in T}. We shall denote it by...

Homework Statement Prove that: The union of a set U and the set of its limit points is the closure of U. Homework Equations Definitions: Closure: The closure of U is the smallest closed set that contains U. Limit points: if z is a limit point in U, then any open circle around...

As a event A\B stands for "A occurs but B does not." Show that the operations of union, intersection and complement can all be expressed using only this operation.A \backslash B = A \cap \bar{B} So far I have resorted to making a truth table with a bunch of A\B combinations that look at A\B...

Homework Statement Suppose A,B,C are sets. Prove that A× (B U C)= (AxB) U (C x A)

1. Suppose open sets V_{\alpha} where V_{\alpha} \subset Y \: \forall \alpha , is it true that the union of all the V_{\alpha} will belong in Y? (i.e. \bigcup_{\alpha} V_{\alpha} \subset Y) Thanks! M

Today I was reading in a probabilities textbook that the probability of the union of two events is: p(E_1 \cup E_2) = p(E_1) + p(E_2) - p(E_1 \cap E_2) and reminded me of the similarity with the dimension of the union of two subspaces of a vector space: dim(V_1 \cup V_2) = dim(V_1) +...