What is Union: Definition and 229 Discussions

Hello! (Wave) I am looking at the proposition: If $(A_n)_{n \in \omega}$ is a sequence of sets and $(f_n)_{n \in \omega}$ is a sequence of functions then: for all $n \in \omega, f_n: \omega \overset{\text{ surjective }}{\rightarrow} A_n$ then there is a function $f: \omega \overset{\text{...

Hello! (Wave) When we have: $f(x)=0, \forall x \in A \wedge f(x)=0, \forall x \in B$, do we conclude that $f(x)=0, \forall x \in A \cap B$ or $f(x)=0, \forall x \in A \cup B$? (Thinking)

Homework Statement Show that, ∪n=2∞[0,1 - 1/n] = [0,1) Homework EquationsThe Attempt at a Solution

Hi! (Smile) I want to prove that for each set $A$: $$A \subset \mathcal P \cup A$$ According to my notes, we prove it like that: Let $x \in A$. We want to show that $x \in \mathcal P \cup A$, so, that: $\exists y \in \mathcal P \cup A$, such that $x=y$. It suffices to show that if $z \in x$...

Hey! :o At any metric space, find a formula that gives the measure of the union of $n$ measurable sets, not necessary disjoint. If the sets are disjoint the measure of the union is $$\mu \left ( \cup_{n=1}^{\infty} A_n \right)=\sum_{n=1}^{\infty}\mu(A_n)$$ right?? And when the sets are not...

Every open sub set of Rp is the union of countable collection of closed sets. I am attaching my attempt as an image file. Please guide me on how I should move ahead. Thank you very much for your help.

Homework Statement Let ##W_1## and ##W_2## be subspaces of a vector space ##V##. Prove that ##W_1 \cup W_2## is a subspace of ##V## if and only if ##W_1 \subseteq W_2## or ##W_2 \subseteq W_1##. Homework Equations A subset ##W## of a vector space ##V## is a subspace of ##V## provided...

What do you guys think about the advantages/disadvantages of being at, say, BofA vs being at a local credit union? For a savings account and for loans, it seems that credit unions take the cake with their low interest rates and (relatively) bigger returns on savings. But for getting a...

Homework Statement In the textbook I'm reading it tells me that A \cup \bigcap B = \bigcap \left\{ A \cup X | X \in B \right\} for B not equal to ø Homework Equations The Attempt at a Solution I don't understand how this would work, the left side of the equation creates a set...

Evening everyone, I have a problem with addition of subspaces. Homework Statement I have to find the dimension of U and dim(V), of the union dim(U+V) and of dim(U\capV) U is spanned by \begin{align} \begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\...

Homework Statement Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A. The Attempt at a Solution The picture I have is if they all intersect (and assuming...

Sixty six cats signed up for the contest MISS CAT 2013. After the first round 21 cats were eliminated because they failed to catch a mouse. Of the remaining cats, 27 had stripes and 32 had one black ear. All striped cats with one black ear got to the final. What is the minimum number of...

By drawing two circles, Mike obtained a figure, which consists of three regions (see picture). At most how many regions could he obtain by drawing two squares? (A) 3 (B) 5 (C) 6 (D) 8 (E) 9

If S is a subset of X, and cS is the complement of S with respect to X, is the union of S and cS equal to X? Seems like a no-brainer but just want to be sure because I've yet to find a book that comments on this.

Homework Statement Consider the following magma, S is not empty; P(S) is the power set. (P(S), U) Now, let A and B be in P(S). What is the GCD of A and B? Homework Equations The Attempt at a Solution If I choose a common divisor of A and B under unions, call it X, I get...

Let ##I## denote the interval ## [0, \infty )## . For each r ## \in I ## define: ##A_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} = r^{2} \}## ##B_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} \leq r^{2} \}## ##C_{r} = \{ ## ... ## : ... > r^{2} \} ## a.)...

Homework Statement Let C=\lbrace(x,y) \in R^2: x^2+y^2=1 \rbrace \cup \lbrace (x,y) \in R^2: (x-1)^2+y^2=1 \rbrace . Give a parameterization of the curve C. The Attempt at a Solution I'm not sure how valid it is but I tried to use a 'piecewise parameterisation', defining it to be...

Hello again! ;) I have also an other question.In order to show that the language $L_{1}=\{w \in \{a,b\}^{*}:w \neq a^{r}b^r, r \geq 0\}$ is context-free, could I use the language $L_{2}=\{w \in \{a,b\}^{*}:w=a^{r}b^{k},r \neq k\} $ ?Isn't it like that:$L_{1}=(\{b\}^{*} \cdot L_{2}) U (L_{2}...

Suppose that $(G,+)$ and $(H,+)$ are both monoids and that the operation $\cdot$ is closed, associative, and distributive over $+$ in $G$ and $H$. My question then is whether or not $(G\cup H,\cdot)$ is necessarily a monoid. I have evidence to suggest that it might, though I cannot prove it.

Hi! I need help for this: Proof equation: (A=B union C and B intersect C=empty set)=>(A\B=C)! Tnx! :o

I recently made a post on Linear and Abstract Algebra and used the following symbol {\bigcup}_{\Omega \subseteq \Gamma , | \Omega | \lt \infty} However, I really wanted (for neatness and clarity) to have the term {\Omega \subseteq \Gamma , | \Omega | \lt \infty} actually under the set...

Homework Statement Let ##G## be a group of order ##n## where ##n## is an odd squarefree prime (that is, ##n=p_1p_2\cdots p_r## where ##p_i## is an odd prime that appears only once, each ##p_i## distinct). Let ##N## be normal in ##G##. If I have that ##|G/N|=p_j## for some prime in the prime...

I intend to show, for a set ##X## containing ##A_i## for all ##i##, $$\overline{\bigcup A_i}\supseteq \bigcup \overline{A_i}.$$ //Proof: We proceed to prove that ##\forall x\in X,~x\in\bigcup\overline{A_i}\implies x\in\overline{\bigcup A_i}##. Equivalently, ##\forall x\in...

Express the set {X E R: (x+3) (7-x) ((x-2)^2) > 0} as a union of intervals

Homework Statement Prove: ##A \cup \varnothing = A## ##A \cap \varnothing = \varnothing## The Attempt at a Solution Intuitively both are true. The first is true because union with nothing will eventually return the original set. The second is true because there is no element that can be in a...

of a countable collection of open intervals. I'm having a hard time seeing how this could be true. For instance, take the open set (0, 10). I'm having a hard time seeing how one could make this into a union of countable open intervals. For instance, (0,1) U (1, 10) or (0, 3) U (3, 6) U (6, 10)...

Problem: Prove that if an element is in the union of two infinite sets then it is not necessarily in their intersection: Proof: Have I solved it correctly?

Problem: Prove that any element in the intersection of two sets is also in their union. I am reading a proof writing book for dummies & the solution given in text is: http://tinypic.com/r/141hn7/5 http://tinypic.com/r/141hn7/5 First Question: In exam/test, is it OK if I write the...

Homework Statement Problem 5 of http://www.math.northwestern.edu/graduate/prelims/anal-f06.pdf Homework Equations The Attempt at a Solution So I've managed to prove it's true if F is an open set. However, I don't know how else to proceed. I tried setting [tex] \mu (F) = m(...

Homework Statement Prove that a finite union of countable sets is also countable. Is an infinite union of countable sets also countable?Homework Equations A set S is countable if and only if there exists an injection from S to N.The Attempt at a Solution I will attempt prove it for the case of...

So I have some friends that attend Cooper Union and they recently informed me about a protest going on there regarding the sudden change to cut the free tuition label. I then saw this yahoo article on the topic...

Homework Statement Let {B_j: j \in J} be an indexed family of sets. Show that \bigcup_{i \in J} B_i \subseteq \bigcap_{j \in J} B_j iff for all i, j, \in J, Bi = Bj. Homework Equations The Attempt at a Solution First show that \bigcup_{i \in J} B_i \subseteq \bigcap_{j...

What is a countable set exactly? HELP? Can someone help guide me through this problem? I'm a bit lost on how to show this... Countable union of countable sets: Let I be a countable set. Let Ai , i ∈ I be a family of sets such that each Ai is countable. We will show that U i ∈ I Ai is countable...

Hi, I am trying to get my head around the Van Kampen Theorem, and how this could be applied to find the fundamental group of X = the union of the unit sphere S2 in R3 and the unit disk in x-y plane? I was thinking of splitting the sphere into 3 regions - two spherical caps each having open...

I am having trouble understanding how the indexed union of ln in the first picture is equal to a subset of the plane; an element of it is a point on one of the lines. If I were to choose say 0 1 2 then the indexed union should be y=0 union y=1 union y=2. These lines would have no points in...

Let $G$ be a group, and $\left \{ H_{i} \right \}_{i\in \mathbb{Z}}$ be an ascending chain of subgroups of $G$; that is, $H_{i}\subseteq H_{j}$ for $i\leqslant j$. Prove that $\bigcup _{i\in \mathbb{Z}}H_{i}$ is a subgroup of $G$. I don't need the proof now. But can you show an example for me...

Homework Statement Can you guys explain to me what the following mean. We are working on probability and unions, and these came up on the homework and need to know what these mean in order to solve the problem. Thanks P(AB) P(AB)c Where c is the compliment. Also i want to...

Homework Statement (i) Let U and V be open subsets of C with a function f defined on U \cup V suppose that both restrictions, f_u \mathrm{and} f_v are continuous. Show that f is continuous. (ii) Illustrate by a specific example that this may not hold if one of the sets U, V is not open...

The following problem was given on a test of mine and I got it completely wrong. If anyone can help me with solving this problem that would be great Let H and K be a subgroup of G, such that H is not equal to G and K is not equal to G . Prove that H union K is not equal to G. Hint: A group...

Homework Statement To give some context, I'm trying to show that \mu(\bigcup^{\infty}_{k=1}A_{k})\leq \sum^{\infty}_{k=1}\mu(A_{k}) where μ is the Lebesgue measure and the A's are a countable set of Borel sets. Since the A's may not be disjoint, I'm trying to rewrite the left side of the...

Pretty much every proof of this I've seen uses the axiom of countable choice at some part or another, and I never got why, since it's pretty cumbersome. Here's the sketch of a proof I wrote for the "fact" that a countable union of countable sets is countable: Let \ P:=\{\pi\in\mathbb{N}|\ \pi \...

I wasn't sure if I should post this in the analysis or topology forum, but this seems to be closely related to compactness so I thought I'd post it here. When dealing with ℝ, the following theorem seems to be really important:"Every non-empty open set G in ℝ can be uniquely expressed as a...

Let G be a finite group and let G = H_1 \cup H_2 \cup H_3 . Show that [G: H_i] = 2 for i = 1, 2, 3. There was a hint for this question saying to first prove that at least one of the subgroups has index 2 in G. So far I am not sure how to even start this problem. I know that the orders of...

The trade balance between EU and China is -156.3€ billions, yet today EU agreed with China (http://uk.reuters.com/article/2012/09/20/uk-eu-china-summit-idUKBRE88J0QR20120920) to avoid trade protectionist measures. They keep doing this because China keeps buying EU countries' bonds and has many...

Homework Statement Let A, C \subseteq ℝn with boundaries B(A) and B(C) respectively. Prove or disprove : B(AUC) O B(A)UB(C) and B(A\capC) O B(A)\capB(C) Where O represents each of these symbols : \subseteq, \supseteq, = Homework Equations I know that double inclusion is going to cut the...

Lets say I have \aleph_1 numbers of sets that each have \aleph_1 number of elements and I want to show that the union of all of these sets has \aleph_1 number of elements. I start with the line segment [0,1] and shift this line segment up by all the reals from 0 to 1. So now...

A first year real analysis textbook presents the following two definitions (where the second builds off the first. (1) Definition (Graph of a map) A and B are sets and f : A \rightarrow B is some map. Then we define the graph of f by G(f) := \{(x,f(x)) \in A \times B : x \in A\}.(2) Other...

why intersection of empty class of sets is the whole space while their union is null set? Book writes that an element will fail to be in the intersection if it fails to be in one of the sets of the class but since there is nothing in the empty class so there is nothing in the empty class that...

Homework Statement If 60% of households subscribe to Metro(M) newspaper, 80% subscribe to local (L) newspaper, and 50% subscribe to both, 1)what's the probability that a random household subscribes to at least one paper? 2) what's the probability that a random household subscribes to...

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