Sets Definition and 1000 Threads

*Sorry wrong section* Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly...

Homework Statement Let x and y be irrational numbers such that x-y is also irrational. Let A={x+r|r is in Q} and B={y+r|r is in Q} Prove that the sets A and B have no elements in common. Homework Equations The Attempt at a Solution Since x and y are in A and B, then...

My problem At temperature 5 degrees the Y=1.00228594 for X=435 and Y=1.000986038 for X=449 and Y=0.999760292 for X=463 At temperature 7 degrees the Y=1.002094781 for X=435 and Y=1.00079709 for X=449 and Y=0.999573015 for X =463 For a new temperature of 9.6 degrees how to find the Y...

Hi Let \( A_1=\mathbb{Z^+} \) and \( \forall n\in \mathbb{Z^+}\) let \( A_{n+1}=\mathcal{P}(A_n) \) I have to come up with an infinite set which is not equinumerous with \( A_n \) for any \( n\in \mathbb{Z^+} \). Clearly \( \mathbb{R}\) will not fit the bill since \( \mathbb{R}\;\sim\; A_2...

Hi Here is the problem. Let A be a set with at least two elements. Also suppose. \[ A\times A \sim A \] Then prove that \[ \mathcal{P}(A)\times \mathcal{P}(A)\sim \mathcal{P}(A) \] Let a and b be the two elements of this set. Then I want to exploit the result that \[ \mathcal{P}(A)\;\sim...

Hi, the question goes as follows: Given two subsets X and Y of a universal set U, prove that: (refer to picture) I'm having particular trouble on D and E, if anyone can clear this up or provide some clarification for me that would be appreciated! I know a venn diagram might not be possible...

I've searched the internet and the forums for any help on this, but I can't seem to find a topic that details what the successive sets will contain. Here is an example question: (I have many HW problems like this, I just don't know where to start) Let L be the language over {a,b} generated by...

Homework Statement The Attempt at a Solution I don't think I'm really understanding this problem. Let me tell you what I know: A set is linearly independent if a_1 A_1 +...+a_n A_n = \vec0 for a_1,...,a_n \in R forces a_1 = ...=a_n = 0. If f,g,h take any of the x_i \in S, then one of the...

I have recently become suspicious of the real numbers. For nearly 3 decades I accepted their axiomatic existence as a complete, ordered archimedian field. The Dedekind-cut, and Cauchy sequence, and "infinite decimal" constructions all made sense to me. And then I started reading about models...

Consider the set S defined recursively as follows: • 3 ∈ S, • if x,y ∈ S,then x−y∈S, • if x∈S, then 2x ∈ S, • S contains no other element. Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ S, ∃ n ∈ Z, x = 3n. What I've got is since 3 is in the set...

Consider the set T defined recursively as follows: • 2∈T, • if x∈T and x>1,then x/2 ∈T, • if x∈T and x>1,then x^2 ∈T, • T contains no other element. Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ T, ∃ n ∈ N, x = 2n. I'm not sure how to...

Homework Statement The a universal set: u = {1,2,3,4,5,6,7,8,9,10} 1) Find the bit string for b = {4,3,3,5,2,3,3,} 2) Find the bit string for the union of two sets. Homework Equations 1)Would I first begin this problem by realizing that set b is the same as {2,3,4,5}? The...

I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets. Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers. For a practical...

hallo I am trying to calculate the probability to obtain 2 sets of linearly independent vectors from a set of binary vectors of length k. For example: k = 4, and therefore I have 2^k = 16 vectors to select from. I want to randomly select 7 vectors (no repetition). What is the...

The rational (and also algebraic) elements of ℂ are closed under addition, multiplication, and rational exponentiation (the algebraic numbers, that is), but not under complex exponentiation. For instance, (-1)^i=e^{-\pi}, with is not rational, and in fact it is even transcendental. Is there any...

& means belong to and # not equal to : $ subsets of A={(t-1,1/t): t&R, t # 0} B= {(x,y) &R^2:y=1/(x+1), x#-1} i started by say A$B let x= t-1 and y=1/t so we have y= 1/(t-1)+ = Y=1/t hence A$B to prove B$A is where i am stuck- as I think I have got my first part wrong anyway and I...

Hello all. Currently working on simplifying some Boolean expressions, one of the questions is: ( A int B U C) int B I do not know how to go about simplifying the first term because there are not any parentheses within it and I have both the intersection and union symbols. Is there an...

Homework Statement Show by example that an infinite union of closed sets in \mathbb{C} need not be closed. The Attempt at a Solution In \mathbb{R} I know that an infinite union of the closed sets A_{n}=[1/n,1-1/n] is open. Not sure if it works in \mathbb{C} as well.

Homework Statement Hi everyone, I have 2 questions that I think are relatively easy but are frustrating in that I just can't seem to get an answer 1st -- If a lateral-to-medial load is applied to the foot, a counteracting moment is produced at the knee joint in a lateral-medial plane, which...

Hi everyone, I was wondering if someone could help with the following: I am doing my undergraduate project and have collected two sets of answers following a survey, which I would like to compare. The questions (29 of them) are mostly Likert style, some allowing for multiple responses...

Homework Statement Find an infinite intersection of open sets in C that is closed. The Attempt at a Solution Consider the sets A_n = (-1/n,1/n). Since 0 in A_n for all n, 0 in \bigcap A_{n}. Here I'm a little stuck -- is the proof in R analogous to the proof in C, or do I need a...

Homework Statement Let E' and E'' be linearly independent sets of vectors in V. Show that E' \cap E'' is linearly independent. The Attempt at a SolutionTo show a contradiction, let E' \cap E'' be linearly dependent. Also let A be all of the vectors in E' \cap E''. Thus, A \subseteq E' and A...

Hello physicsforum - I recently began self-studying real analysis on a whim and I have run into some trouble understanding the idea of compactness. I have no accessible living sources near me and I have yet to find a source that has explained the matter clearly, so I shall turn here for...

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...

See Attachment 2 for question or read below Describe and compare the solution sets of X_1 + 5 X_2 - 3 X_3 = 0 and X_1 + 5 X_2 - 3 X_3 = -2. See Attachment 1 for answer from back of book I do not understand how the answer in the back of the book answers the question or were to even begin to get...

Suppose we have non-empty A_{1} and non-empty A_{2} which are both open. By "open" I mean all points of A_{1} and A_{2} are internal points. There is an argument -- which I have seen online and in textbooks -- that A_{1} \cap A_{2} = A is open (assuming A is non-empty) since: 1. For some x...

So this pony was up for sale it only costs a monkey, so i went to see it, it was a little beauty so i agreed to buy it, i asked how much for the tack, the guy said a pony so i bought that as well. Then i saw the other pony and asked the guy how much, he said the same a monkey, so i bought...

So this was an exam question that our professor handed out ( In class. I didn't get the question right) Let E be a subset of R^n, n>= 2. Suppose that E measurable and m(E)>0. Prove that: E+E = {x+y: x in E, y in E } contains an open ball. (The text Zygmund that we used showed an...

Homework Statement Show that the two sets of vectors {A=(1,1,0), B=(0,0,1)} and {C=(1,1,1), D=(-1,-1,1)} span the same subspace of R3. Homework Equations {A=(1,1,0), B=(0,0,1)} {C=(1,1,1), D=(-1,-1,1)} The Attempt at a Solution aA+bB=(a,a,0)+(0,0,b)=(a,a,b)...

Homework Statement The set of all pairs of real numbers of the form (1,x) with the operations: (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar Is this a vector space?Homework Equations (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)The Attempt at a Solution I verified most of the axioms...

I'm working on some topology in \mathbb{R}^n problem, and I run across this: Given \{F_n\} a family of subsets of \mathbb{R}^n , then if x is a point in the clausure of the union of the family, then x \in \overline{\cup F_n} wich means that for every \delta > 0 one has B(x,\delta) \cap...

I have looked it in the Wikipedia, but no simple example. So I am not sure. Is the indexed family of sets just power sets, indexed (indexing means labeling as I understand)? For example the indexed family of sets of set A ={1,2,3,4,5,6} is just the collection of element from power set. A sub 1...

Homework Statement If [n] and [m] are equal, then they are bijective correspondent. I define f \subset\{(n,m)\mid n \in [n], m\in [m]\}. Suppose [n]=[m]. Let(n,m_1),(n,m_2)\in f. Because [n]=[m], then m_1=m_2. So for all n \in [n], there exists a unique m\in [m] such that f(n)=m. So f...

Let A = {(x,y) in R^2 | x^2 + y^2 <= 81} Let B = {((x,y) in R^2 | (x-10)^2 + (y-10)^2 <= 1} then here "A intersection B" is the empty set. Then let x_n be the sequence (0,10-(2/n)) which is a sequence in A and y_n be the sequence (10/n,10) which is a sequence in B. would |x_n - y_n| tend...

Homework Statement Let {xn} be a sequence of real numbers. Let E denote the set of all numbers z that have the property that there exists a subsequence {xnk} convergent to z. Show that E is closed. Homework Equations A closed set must contain all of its accumulation points. Sets with no...

Homework Statement a) If U and V are open sets, then the intersection of U and V (written U \cap V) is an open set. b) Is this true for an infinite collection of open sets? Homework Equations Just knowledge about open sets. The Attempt at a Solution a) Let U and V be open...

I've been reading a book called Superfractals, and I'm having trouble with a particular proof: Definitions: The distance from a point x \in X to a set B \in \mathbb{H}(X) (where \mathbb{H}(X) is the space of nonempty compact subsets of X is: D_B(x):=\mbox{min}\lbrace d(x,b):b \in B\rbrace The...

I've got a Mathematica question which might be quite basic, but I couldn't find much about it in the documentation (possibly because it's so basic) so please bear with me! I have a set of data, call it xi(ρ), which I want to integrate over some distribution function (log-normal in this case)...

Homework Statement Consider a set of vectors: S = {v_{1}, v_{2}, v_{3}, v_{4}\subset ℝ^{3} a) Can S be a spanning set for ℝ^{3}? Give reasons for your answer. b) Will all such sets S be spanning sets? Give a reason for your answer. The Attempt at a Solution a) Yes, because a...

Are there any down to Earth examples besides the empty set? Edit: No discrete metric shenanigans either.

In Rudin's Principles of Mathematical Analysis, Theorem 2.43 is that all nonempty perfect sets in R^k are uncountable. The proof Rudin gives goes like this: Let P be a nonempty perfect set in R^k. Since P has limit points, P is infinite. Suppose P is countable and denote the points of P by...

Hi everyone. My first post on this great forum, keep up all the good ideas. Apologies if this is in the wrong section and for any lack of appropriate jargon in my post. I am not a mathematician. I have a theory / lemma which I would like your feedback on:- Take a finite set S of integers which...

Homework Statement Do the polynomials t^{3} + 2t + 1,t^{2} - t + 2, t^{3} +2, -t^{3} + t^{2} - 5t + 2 span P_{3}? Homework Equations N/A The Attempt at a Solution My attempt: let at^{3} + bt^{2} + ct + d be an arbitrary vector in P_{3}, then: c_{1}(t^{3} + 2t + 1) + c_{2}(t^{2} -...

Homework Statement Using the fact that a set S is linearly dependent if and only if at least one of the vectors, vj, can be expressed as a linear combination of the remaining vectors, obtain necessary and sufficient conditions for a set {u,v} of 2 vectors to be linearly independent. Determine...

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?

Russell's paradox concerns itself with the set S=\{x|x\notin x\}\;\;\;S\in S ? but it is supposedly solved in ZFC theory. Now, what about the set U=\{y|y\in y\}\;\;\;U\in U ? Is U an element of itself?

Do tuples exist which aren't elements of a cartesian product of sets? Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)

I have difficulty understanding the following Theorem If U is open in ℝ^2, F: U \rightarrow ℝ is a differentiable function with Lipschitz derivative, and X_c=\{x\in U|F(x)=c\}, then X_c is a smooth curve if [\operatorname{D}F(\textbf{a})] is onto for \textbf{a}\in X_c; i.e., if \big[...

Ok, so basically I am trying to decide whether my mathematics is valid or if there is some subtly which I am missing: Lets say I have a 1-1 strictly increasing point-wise continuous function f: R -> R, and I want to show that the inverse function g: f(R) -> R is also point-wise continuous...