Union Definition and 229 Threads

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) +...

Homework Statement This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach. Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t} where Y is a random variable and t is any real number. Fix a real number...

Homework Statement What does it mean to have a union of two subsets? Could someone provide me with an example. Thank you.

Eh, kind of stuck on this question. I need some suggestions on how to tackle the problem.. Homework Statement Let U and V be the subspaces of R_3 defined by: U = {x: aT * x = 0} and V = {x: bT * x = 0} (T means transpose) where a = [1; 1; 0] and b = [0; 1; -1] Demonstrate that...

I was trying to prove that the sigma algebra generated by the set of open intervals is the same as the sigma algebra generated by the set of open sets. This proof devolves into proving the statement in the title. I think rational numbers must be brought into the picture to prove this stmt but I...

well, well, well... America---the future---what do you think and like/dislike about the speech? (Why are the Republicans all dressed alike and most clap only at certain times?)

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...

Homework Statement Let (X,d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only if it can be written as the union of a family of open balls of the form Br(x) = {y ∈ X|d(x,y) < r} (the radius r may depend on the point x). Homework Equations...

If x1, x2 positive random variables and we have the following two events: A={x1 > δ} B={x2> k-δ} where 0<δ<k then is it true that: P(A U B) = P( x1+x2 > δ+(k-δ)=k ) ? If true can you explain why is that? Thank you

In R, every nonempty open set is the disjoint union of a countable collection of open intervals. (Royden/Fitzpatrick, 4th edition) What is the most general setting in which every open set is a disjoint union of countable collection of open balls (or bases)? In R^n? In metric spaces? In second...

Suppose that a union's goal is to maximize the total wage income received by union workers, namely, the average union wage times the number of union workers employed. To achieve this goal, the union should: A. Decrease the union wage rate if labor demand is inelastic and increase the wage...

Homework Statement I know this is probably fairly trivial, but for the life of me I cannot remember or reconstruct the proof for the proposition, "The sum of the lengths of a finite number of overlapping open intervals is greater than the length of their union." Homework Equations Not...

Homework Statement Prove that any open subset of \Real can be written as an at most countable union of disjoint open intervals. Homework Equations An at most countable set is either finite or infinitely countable. The Attempt at a Solution It seems very intuitive but I am at lost...

One of the exercises in the text I'm using for self-study asks to prove that the union of a pair of atlases A and B on a manifold is another atlas. However, I don't see any way to show that two charts C,D in A\cup B with C\in A~,~D\in B are compatible. Could anyone give me a bit of help? Maybe...

Homework Statement Toss a coin 3 times. What is the probability that we get a head on the first toss or a head on the second toss or a head on the third toss? Homework Equations Pr(AorB)=Pr(A)+Pr(B)-Pr(AandB) The Attempt at a Solution A=head on 1st toss B=head on 2nd toss...

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...

Hi all, I'm getting stuck on this problem. Homework Statement I am asked to show that that the open ball in the plane {|x|} < 1} can be written as a countable union of rectangles [a_1, a_2] x [b_1,b_2], but the closed ball in the plane {|x| <= 1} cannot be written as a countable union of...

I don't quite understand the meaning of "infinite union" and "infinite intersection". Is an infinite union ∞ U Ak k=1 being defined as a limit lim (A1 U A2 U ... U An) ? n->∞ How about an infinite intersection? Thanks!

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}.

Homework Statement Show that if A1, A2, ..., An are independent events then P(A1 U A2 U ... An) = 1 - [1-P(A1)][1-P(A2)]...[1-P(An)] Homework Equations If A and B are independent then the probability of their intersection is P(A)P(B). The same can also be said of AC and B. The...

Hi, I have four similar problems that I am not sure how to do: Given: A1 and A2 are in X, B1 and B2 are in Y f: X->Y, g - inverse of f I have to either prove or if false find counterargument 1. f(A1 U A2) = f(A1) U f(A2) 2. f(A1 n A2) = f(A1) n f(A2) 3. g(-1)(B1 U B2) = g(B1) U g(B2) 4...

Soviet Union, where'd it go wrong? Hi all, a few days ago i was thinking about how China is Communist state and i wondered how long it may take for China to become democratic, then i remeabered how Russia used to be a Communist state and eventually became democratic. I tried to think of how...

This seems like a simple problem but I cannot find an answer. Imagine I have 2 bags of samples with estimated means \hat{x} and \hat{y} and estimated variances \hat{\sigma_x} and \hat{\sigma_y}. The bags contain n and m samples respectivelly. Now assume I mix all the samples in a single...

Why is this true: (A - B) union (B- A) = (A union B) - (A intersection B) wouldn't the union of A and B everything that is in A or B? And since A - B and B - A don't contain any elements from the other set, wouldn't the union of these be equal to union of A and B? So wouldn't it make sense...

Homework Statement Given a set A \in R^m, B_n \in R^m for n \in N, show that A \ Union {from n = 1 to inf} B_n = Intersection {from n = 1 to inf} (A \ B_n} Homework Equations Same equation as above The Attempt at a Solution I think I have a solution in mind, but I wanted to...

I have a program to define union and intersection on 2 sets. I know I need 2 different input files, but other than that, I'm clueless.

Is the infinite union of uncountable sets also uncountable? Just need a yes or no. Thanks.

Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help

Let A, B, and A\alpha denote subsets of a space X. neighborhood of \bigcupA\alpha \supset \bigcup neighborhood of A\alpha; give an example where equality fails.Criticize the following "proof" of the above statement: if {A\alpha} is a collection of sets in X and if x \in neighborhood of...

I'm sick of the greed that comes w/ having my money in a big bank. Anyone have their money in a credit union? Is convenience a problem?

Homework Statement Simplify the expression: (B union C) intersection (B union NOT-C) intersection (NOT-B union C) The Attempt at a Solution I have no clue how to attempt this question, as every time I do attempt it I get a different solution.

It seems intuitive that the power set of a union of sets P(XunionY) is not a subset of the union of the two respective power sets P(X)unionP(Y). For finite sets the former will have more elements than the latter. However, I can't figure out what is wrong with the following line of reasoning...

Homework Statement let Ai be a subset of the reals and i is element of I = (1,...,n) now let A = UNION i is element of I Ai show that sup(A) = supiEI(sup(A)) Homework Equations The Attempt at a Solution My idea of solving was taking the limits of both sides but I'm...

Homework Statement Prove that the union of intervals [1,n] from n=1 to n=infinity is all of N. The Attempt at a Solution Do I use induction on this? Archimedes? (This question is before the section of Archimedes though). I need help on how to start it!

Homework Statement Given A1 superset of A2 superset of A3 superset of A4 ... and so on how can i construct sets B1, B2, ... so that each Bi's are disjoint. The goal is to get the infinite intersection of Ai = the infinite union of BiHomework Equations De morgans law: (AUB)^c = (A^c N B^c)...

Homework Statement Let V be a vector space over an infinite field. Prove that V is not the union of finitely many proper subspaces of V. The attempt at a solution Suppose V is the union of the proper subspaces U1, ..., Un. Let ui be a vector not in Ui. If u1 + ... + un is in the union...

Prove that if (H,o) and (K,o) are subgroups of a group (G,o), then (H \cap K,o) is a subgroup of (G,o). Proof: The identity e of G is in H and K, so e \in H\capK and H\capK is not empty. Assume j,k \in H\capK. Thus jk^{-1} is in H and K, since j and k are in H and K. Therefore, jk^{-1}...

I've read in a couple books (e.g. in Lee Smolin's three roads to quantum gravity) about the possible union of LQG and string theory. In other words, that string theory and LQG can possibly be reformulated into a single TOE. I've been unable to find anything more on this idea through google and...

Homework Statement I solve the equation of one function, which comes out with two solutions: 1. cosx=-1, x=(2k+1)\pi ; 2. cosx=1, x=2k\pi (k \in \mathbb{Z}) Homework EquationsThe Attempt at a Solution Now, we need to find union between the two of the solutions: {\pi + 2k\pi}\cup{2k\pi...

Homework Statement Prove (A is a union of B)/(A is an intersection of B)=(A/B) is a union of (B/A) Homework Equations The Attempt at a Solution Could someone first help me translate all of this into plain English. I don't really understand what I need to prove. Would I start off...

Homework Statement Prove that if A_1,A_2,…,A_n and B are sets, then (A_1 – B) U (A_2 – B) U … U (A_n – B) = (A_1 U A_2 U … U A_n) – B. Homework Equations The chapter this is in is based on mathematical induction, which might be a big hint. Mathematical induction: Step 1: Prove for the...

Homework Statement If X1, X2, X3, ... Xk are independent events, prove that P(X1 U X2 U X3 U ... U Xk) = 1 - [1 - P(X1)][1-P(X2)]...[1-P(Xk)] Homework Equations The Attempt at a Solution Well I have tried a few methods, but I know it's got something to do with P(X1) = cc(P(X1))...

In my experience, whip provides the most speed/spin in table tennis...you don't have to be a tt pro to answer this question though... Now, the issue is, inorder to whip, we need to keep our muscles relaxed...meaning, to sprint, our leg muscles, to swing, our arm muscles and torso...etc...the...