Subsets Definition and 209 Threads

I have trouble visualizing what exactly these are. Vector Space, Subset, Sub Space... What's the difference and how can I "see" it. I'm a very visual person.

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...

Homework Statement T = {(1,1,1),(0,0,1)} is a subset of R^{3} but not a subspace sol i have to prove it holds for addition and scalar multiplication so let x=(1,1,1) and y =(0,0,1) so x+y = (1,1,2) so it holds let \alpha = a scalar then \alphax = (\alpha,\alpha,\alpha)...

Homework Statement Let G be a group acting on a set X, and let g in G. Show that a subset Y of X is invariant under the action of the subgroup <g> of G iff gY=Y. When Y is finite, show that assuming gY is a subset of Y is enough. Homework Equations If Y is a subset of X, we write GY for...

Homework Statement Prove or disprove that for any non empty sets X and Y, any subset A of X, and f:X->Y, f(f^-1(A))=A Homework Equations The Attempt at a Solution I know that for x element of A f(x) = f(A) so x is an element of f^-1(f(A)). If I can assume f is injective then I can go from...

Homework Statement H = {(x,y,z) \in R^3 | x + y^2 + z = 0} \subseteq R^3 T = {A \in M2,2 | AT = A} \subseteq M2,2 The Attempt at a Solution Our lecturer wasn't quite clear about how to go about this. He talked out closed under addition and multiplication but that's about it...

Homework Statement General Case: Let f:A \rightarrow B be a function. Show that f( \bigcap_{\alpha\in\Lambda} T_{\alpha}) = \bigcap_{\alpha\in\Lambda} f(T_{\alpha}) for all choices of \{T_{\alpha}\}_{\alpha\in\Lambda if and only if f is one-to-one. Simpler Case: Let f:A \rightarrow B...

Hi, I'm reading HL Royden's real analysis, though my question pertains more to set theory. Let X be a set. Then the intersection of an empty collection of subsets of X is equal to X. I understand this is not an intersection of empty subsets but it is still very counter-intuitive. Can...

Here we go...wheeeee Homework Statement For each of the following subsets of F3, determine whether it is a subspace of F3 (a) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1+2x_2+3x_3=0} (b) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1+2x_2+3x_3=4} (c) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1x_2x_3=0} (d)...

Let (X,d) be a metric space A and B nonempty subsets of X and A is open. Show: A\capB = \oslash Iff A\capB(closure)= empty Only B closure it is easy to show rigth to left but how can i use A's open property I try to solve with contradiction s.t. there exist r>0 Br(p)\subseteqA\capB(closure) but...

I am beginning to study set theory and came across the following example: Let \mathcal{A} be the empty family of subsets of \mathbb{R}. Since \mathcal{A} is empty, every member of \mathcal{A} contains all real numbers. That is, ((\forall A)(A\in\mathcal{A}\Rightarrow x\in A)) is true for...

i have the following problem that i can't figure out. i have a set ID's which i pseudo-randomly split into 2 subsets A and B. let's say for the sake of simplicity i did it twice only, so i have subsets A&B and A`&B`. the sizes in the different splittings are the same - not sizes of A and B...

Prove that if s1 and s2 are subsets of a vectorspaceV such that... Homework Statement Prove that if s1 and s2 are subsets of a vector space V such that S1 is a subset of S2,then span(S1) is a subset of span(s2). In particular, if s1 is a subset of s2 and span(s1)=V, deduce that span(s2)=V...

Assignment question: Let V = P (R) and for j >= 1 define T_j(f(x)) = f^j (x) where f^j(x) is the jth derivative of f(x). Prove that the set {T_1, T_2,..., T_n } is a linearly independent subset of L(V) for any positive integer n. I have no idea how...

Homework Statement Which of the following subsets of the vector space R^R of all functions from R to R are subspaces? (proofs or counterexamples required) U:= f R^R, f is differentiable and f'(0) = 0 V:= fR^R, f is polynomial of the form f=at^2 for some aR = There exists a of the set...

I had a quick question: Is the following proof of the theorem below correct? Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1. Proof: Since C is convex, then t*x + (1-t)*y...

Consider the closed interval A = [a,b]\subset \mathbb{R}. Are the only dense subsets of A the set of all rational numbers in A and the set of all irrational numbers in A? Something tells me that there's got to be more than that, but I can't think of any examples. Thanks in advance for your help.

Homework Statement Let A be an infinite set which is not countable and let B \subset A be a countably infinite set. (1) Show that A - B is also infinite and not countable (2) Show that A and A - B have the same cardinality Homework Equations Hints written directly: "Show that...

Homework Statement 1) Let S be a set and p: SxS->S be a binary operation. If T is a subset of S, then T is closed under p if p: TxT->T. As an example let S = integers and T be even Integers, and p be ordinary addition. Under which operations +,-,*,/ is the set Q closed? Under which...

Show that the following statements are equivalent for any topological space (X, \tau). (a) Whenever A, B are mutually separated subsets of X, there exist open disjoint U, V such that A \subseteq U and B \subseteq V. (b) (X, \tau) is hereditarily normal. Background: Definition- Sets H...

Homework Statement a) construct a subset of two-dimensional space closed under vector addition and even subtraction, but not under scalar multiplication. b) construct a subset of two-dimensional space (other than two opposite quadrants) closed under scalar multiplication but not under...

Alright, 2+1/2 problems here: First: 36. Let F be a function from the set A to the set B. Let S and T be subsets of A. Show that: a) f(S\cupT)=f(S)\cupf(T) b)f(S\capT)\subseteq f(S)\capf(T) Note: This must be done using a membership proof. i.e. prove that...

Homework Statement Let I = <f(x)>, J =<g(x)> be ideals in F[x]. prove that I\subsetJ \leftrightarrow g(x)|f(x) Homework Equations The Attempt at a Solution If I is a subset of J then does that mean that f is in J also and by definition of an ideal g*some b in J must equal something...

This may be a stupid question, but I just confused myself on compactness. For some reason I can't convince myself that ANY subset of a compact set isn't compact in general; just closed subsets. Suppose K is a compact set and F \subset K. Then if (V_{\alpha}) is an open cover of K, K \subset...

Give an example of a closed set S in R^2 such that the closure of the interior of S does not equal to S (in set notation). I have no idea where to start...any help would be nice! Thanks!

The question: How many ways are there to choose three subsets, A, B, and C of [n] that satisfy A\subseteqB\subseteqC? My attempt: Since C is the primary subset of [n] (the set upon which A and B are built) then the maximum number of ways to choose that set is based upon both n and |C| so...

Homework Statement Let f: A \rightarrow B be given and let {X_{\alpha}} for \alpha \in I be an indexed family of subsets of A. Prove: a) f(U_{\alpha\inI} X_{\alpha}) = U_{\alpha\inI}f(X_{\alpha}) The Attempt at a Solution To prove these two things are equal I must show...

How do I prove that the set of all finite subsets of an infinite set has the same cardinality as that infinite set?

So if A is an infinite set, we know that |A|+|A|=|A|. But are we allowed to go backwards, i.e. divide A into two disjoint subsets B and C such that A = B U C, and |B|=|C|=|A|. For the integers and for the reals, this is clear, e.g. R = (-infinity,0) U [0, infinity), each with cardinality c...

Proof subset? Given three sets A, B, and C, set X = (A-B) U (B-C) U (C-A) and Y = (A∩B∩C) complement C. Prove that X is subset of Y. Is Y necessarily a subset of X? If yes, prove it. If no, why? ---When I draw the two venn diagrams X and Y, they are the same, but I don't know how to prove...

Homework Statement How many subsets of a set with 100 elements have more than one element? The thing throwing me off here is the zero-set, and whether it counts as an element or that. Can someone start me off?

Homework Statement Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X\subseteqY. Show that Y\cap(X+Z) = X + (Y\capZ). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.) Homework Equations The Attempt at a...

Homework Statement Which of the following subsets of the vector space R^R of all functions from R to R are subspaces? (proofs or counterexamples required) U:= f \inR^R, f is differentiable and f'(0) = 0 V:= f\inR^R, f is polynomial of the form f=at^2 for some a\inR = There exists a...

Homework Statement Let U be a non-empty, convex, open subset of R^2. Prove that U is homeomorphic to R^2. Hint: First prove that the intersection of a line in R^2 with U (if non-empty) is homeomorphic to an open interval in R^1. Then use radial projections.Homework Equations We just have the...

Homework Statement Conjecture: If K=a union of subsets of G with K open then each subset in the union is open The Attempt at a Solution Can't really see the proof. In fact it's false as any non discrete topology have open sets which are a union of subsets whch may not be open.

Homework Statement Prove that the set of all subsets of the natural numbers is uncountable. Homework Equations All of the countability stuff - including Cantor's diagonal argument The Attempt at a Solution I think I have this one figured out, but I was wondering if somebody would...

Homework Statement The Premise: Here One must prove that that R^n and Ø are the two subsets of R^n, which is both open and closed. You must that these are the only subsets of R^n which has this property! Let X \subseteq \mathbb{R}^n be a subset, which is both open and close, and here...

Prove that an infinite Hausdorff space has an infinite collection of mutually disjoint open subsets.

Homework Statement What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2? R, in this question, is the real numbers. Homework Equations I know that, for example, V={(0,0)} is a subset for R^2 that...

Springfield Football Club plan to field a team of 3 forwards, 4 mid-fielders and 3 defenders and a goalkeeper. Assuming they have 8 forwards, 6 mid-fielders, 5 defenders and 2 goal- keepers on their books how many teams can they make? i tried doing: (8C3) x (6C4) x (5C3) x (2C1) but...

Homework Statement X=R real numbers, U in T, the topology <=> U is a subset of R and for each s in U there is a t>s such that [s,t) is a subset of U where [s,t) = {x in R; s<=x<t} Find the closure of each of the subsets of X: (a,b), [a,b), (a,b], [a,b] The Attempt at a Solution I don't...

* If p1,p2,……pm span Pn, write down a mathematical relationship between m and n. I know that Pn means the space of all polynomials of degree at most n, and this is an (n+1) dimension space, but I am not sure what kind of mathematical relationship the question is looking for :s Any help is...

I am having some troubles understanding the following, any help to me will be greatly appreciated. 1) Let S1 = {x E R^n | f(x)>0 or =0} Let S2 = {x E R^n | f(x)=0} Both sets S1 and S2 are "closed" >>>>>I understand why S1 is closed, but I don't get why S2 is closed, can anyone...

Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...

Permutations of subsets with like objects Question Hello, I'd like to know if I solved the question correctly; if not, I'd appreciate some help. Question: Calculate the number of permutations for a subset of 3 objects from a superset of 8 objects where 5 are alike. My solution attempt...

Homework Statement Let: \alpha \in C_{4}[x] (the space of all 4-deg complex ploymonials) We'll define: U = Sp( \{\ x^3 - 2i \alpha x, x^2 +1 \} ) \\ W = Sp( \{\ x^3 + ix, (1-i)x^2 - \alpha x \}) as subspaces of C_{4}[x] a) find all values of alpha so that: W \cap U \neq \{ 0 \}...

How can I prove that \left| {\left\{ {A \subset \mathbb{N}:\left| A \right| \in \mathbb{N}} \right\}} \right| = \left| \mathbb{N} \right| ?

Hello everyone I'm confused on how he got this... the question says: Let S = {a,b,c} and for each integer i = 0, 1, 2, 3, let S_i be the set of all subests of S that have i elements. List the elements in S_o,S_1,S_2,S_3. Is {S_0,S_1,S_2,S_3} a partion of P(S)? Here is the answer...

Hello everyone. There arn't any problems like this in this section, so I'm kind of lost on what they want. It says... Let S = {a,b,c} and for each integer i = 0, 1, 2, 3 let S_i be the set of all subsets of S that have i elements. List the elemnts in S_o,S_1,S_2,S_3. Is {S_0,S_1,S_2,S_3}...

Can someone confirm/disprove the following: If X\subset\mathbb{P} is infinite, then \sum_{n\in X}\frac{1}{n} diverges or is irrational.