Set Definition and 1000 Threads

Homework Statement A= {a1,...am}, B= {b1,...an}. If f: A→B is a function, then f(a1) can take anyone of the n values b1,...bn. Similarly f(a2). Then there are nm such function. I understand this part. So in my book, using this principle, nC0 + nC1 + ... + nCn = 2n is proved. It has taken a...

Homework Statement [/B] I am going through Apostol's Calculus volume 1 and am working through I 2.5 #3. I'm not very familiar with doing proofs so I just wanted to make sure that I got the right idea here. Here's the question: Let A = {1}, B = {1,2} Prove: 1. ## A \subset B ## 2. ## A...

Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...

A have the set consisting of the complex numbers ##1 + 3r \cos \theta - i r \sin \theta##, where ## r \in [0,1]## and ##\theta## may vary between ##0## and ##2 \pi##. This is my first encounter with mathematica, and am having difficulty discerning between the methods I have found online which...

Hey friends, My college team takes part in college level racing events. They currently use a KTM 390 engine(race limit is 600). WIth the 20mm restrictor rule we can generate around 30-33hp after some tuning, after the restriction of course. Now we want more power. We need it. So we have 2...

Following is from Wolfram Mathworld "A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: The empty set is in T.X is in T.The intersection of a finite number of sets in T is also in T.The...

Hello! (Wave) An independent set of a graph $G=(V,E)$ is a subset $V' \subseteq V$ of vertices such that each edge in $E$ is incident on at most one vertex in $V'$. The independent-set problem is to find a maximum-size independent set in $G$. Formulate a related decision problem for the...

I have done basic experiments where a laser is shot thru a grating and a lens is inserted at the diffraction pattern to put the spectra back together and form an image of the grating. apart from this being the single most mind blowing experiment ever to witness I have some questions about the...

Hello, I'm having a little trouble figuring out the following problem: Consider the set of number a, 2a, 3a, ..., na where a and n are positive integers. (i) Show that the expression for the mean of this set is \frac{a(n+1)}{2}. So far the only work I've been able to muster up is: Mean =...

This is a rather simple question, so it has been rattling my brain recently. Consider a surjective map ## f : S \rightarrow T ## where both ## S ## and ## T ## are finite sets of equal cardinality. Then is ## f ## necessarily injective? I proved the converse, which turned out to be quite...

Homework Statement Find the truth set of the given equivalence. Assume U=ℝ #56. (x2=1)↔[(x=1)∨(x=-1)] Source: Principles of Mathematics by Allendoefer and Oakley section 1.10 Homework Equations {x I px↔qx}=(P∩Q)∪(P'∩Q')[/B] The Attempt at a Solution P={x I px}={x I x2=1}={x I x=1 or -1)...

1. Find P'={x I ~px} for the given open sentences px. #25. x2≥4. (Problem from 1.10, Principles of Mathematics by Allendoerfer and Oakley. Solution offered at the back of the book: {x I -4<x<4}.Homework Equations If P={x∈ℝ I px} then P'={x∈ℝ I ~px}[/B]The Attempt at a Solution x2≥4 ⇒ x≤-2 or...

Homework Statement 1. Prove that if A \cap B = A and A \cup B = A , then A = B 2. Show that in general (A-B) \cup B \neq A 3. Prove that (A-B) \cap C = (A \cap C) - (B \cap C) 4. Prove that \cup_{\alpha} A_{\alpha} - \cup_{\alpha} B_{\alpha} \subset \cup_{\alpha} (A_{\alpha} -...

Homework Statement We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets...

Is it possible that there is a set of postulates (or statements) whose logical implications imply the quantum nature of our Universe, the postulates of quantum mechanics? Could there be more than one set? Thanks for any help!

OK, this is embarrassing, but I never looked carefully at this elementary point. We say that if p implies q P is the set of all things for which p is true Q is the set of all things for which q is true then Q ⊆ P. Also that the set of all things for which p&q is true equals P∩Q But p & q...

Homework Statement Find the solution set of 2^(2x-2)-2*2^(x-1)=8 Homework EquationsThe Attempt at a Solution 2^(2x-2)-2*2^(x-1)=8 2^(2x-2)-2^(x-1+1)=8 2^(2x-2)-2^(x)=8 2^(2x-2)-2^(x)-8=0 2^(2(x-1))-2^(x)-8=0 I cannot solve the equation, I just need direction with this step and will attempt...

Homework Statement Hi everyone. I'm teaching introductory chemistry this semester, but I don't have much of a chem background. We are about to start acid base titrations. Does the standard solution of known concentration go in the buret tube or the erlenmeyer flask? Does it matter? Homework...

I have two a convex set: {(x1, x2): 1≤ ∣x1∣ ≤2, ∣x2−3∣ ≤ 2} I have to find two points in the set for which the line segment joining the points goes outside the set. I have graphed the function and found my convex set. My question is, how do I find these two points? I have found various points...

So I just finished "Book of Proof" and I'm looking for a more rigourous book on axiomatic set theory, including Gödel's theorems.Any recommendations?

Homework Statement The largest value of r for which the region represented by the set { ω ε C / |ω - 4 - i| ≤ r} is contained in the region represented by the set { z ε C / |z - 1| ≤ |z + i|}, is equal to : √17 2√2 3/2 √2 5/2 √2 Homework Equations complex number = a + ib where a,b ε R The...

Can anyone explain why sum of Ft has been set to be = 0? The slope is smooth so no frictional forces act but by setting it to 0 they are neglecting the component of the weight along the slope? Having said that, this is a curved slope and not a straight one so does the component of the weight...

I'm currently an undergrad in math who's going to graduate next year. I'm interested in doing research in set theory (not now of course, perhaps in grad school). Unfortunately, I'm at a liberal arts school and there are no set theorists in the math department. All they offer is a naiive set...

Wiki says "A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context." To me the term (linear) Vector Space has always seemed a little mysterious ... how far wrong would I be in thinking of a vector...

Homework Statement If I have a subclass that implements a set class how can I avoid having to override all the set classes methods? Homework Equations None The Attempt at a Solution I have an interface: import java.util.Set; /** * An extended Set where there is added...

Homework Statement Second try at the proof, this time with correct vocabulary, I hope Prove that an open ball is an open set. Homework EquationsThe Attempt at a Solution Let B(P_0, r) be an open ball in \mathbb{R}^m, where P_0 is the centerpoint of the ball and r > 0 is its radius. Assume...

Homework Statement Prove that an open sphere in \mathbb{R}^m is an open set. Homework EquationsThe Attempt at a SolutionTo show that an open sphere is an open set, any point inside the sphere has to be an interior point: Let us have a sphere B(P_0, r), r > 0, where P_0 is the centerpoint and r...

Hi, I know this is a basic question however, I am seeking absolute verification on these two points. 1) Quantum Mechanics is a set of Laws. 2) Laws only describe what we see. Theories give us the reason behind them.

Hello there, I am just starting quantum physics with the textbook by griffiths. My lecturer has told me that the set of functions representing stationary states in Hilbert space forms an orthogonal set. He was however unable to prove it. Furthermore he said that it is not always the case, but...

I am reading Munkres book, "Topology" (Second Edition). I need help with an aspect of Theorem 18.2 Part (f) concerning the inverse image of a set under the restriction of a function ... Theorem 18.2 Part f reads as follows: In the above text we read: " ... ... Let $$V$$ be an open set in...

So I was helping my sister on homework and there was this problem: 2 abs(2x + 4) +1 > or equal to -3 teacher told her to ignore the -3 and just set it equal to zero. Soo should you? This question got me confused. can't you just go about solving, bringing the 1 to the left and then dividing by 2...

I should have studies statistics in school. I have been collecting some trip data from you commute to and from work, and I want to determine what factors are influencing trip duration. There are multiple factors, which I cannot easily isolate. The factors I'm most interested in are: - duration...

I have seen many establishments having a backup electric supply using a Diesel Generator Set (AC / 3 phase). e.g. Something like in the Photo Below. What happens if the connected load grows beyond the capacity of a single generator? e.g. The unit in the photo is rated for 200 kVA. Suppose the...

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...

In an earlier post here I wanted to chop up a three-sphere into cubes, Ben suspected it was not possible and I have no reason to think otherwise. From earlier help by Fezro, here, I may be able to move this forward. Assuming the posts by Fezro are correct I think I can come up with a set of...

Lately I have been attempting (and failing miserably at) whatever sample Putnam questions I can find on the internet. Here is my latest endeavor. I found this question on the Kansas State University website, so I think I am allowed to post it. I must warn you that I know almost nothing about...

So I'm reading Naive Set Theory by Paul Halmos. He asks: His response is that no ##x## fails to meet the requirements, thus, all ##x##es do. He reasons that if it is not true for a given ##x## that ##x \in X~ \mathrm{for ~ every} ~X~ \mathrm{in} ~ \phi##, then there must exist an ##X## in...

Can someone explain me,why Wheatstone bridge is most sensitive when all four resistances say A,B,C and D are equal?as far as i know condition for Wheatstone Bridge is A/B=C/D.

So I'm reading up on some set theory, and I came to the axiom of pairing. The book uses that axiom to prove/define a set which contains the elements of two sets and only the elements of those two sets. ##~~B## is the set which contains the elements and only the elements of sets ##a## and...

Let $A$ be the set of all positive integers $a$ such that $2^{2008}+2^a+1$ is a square. Find the smallest number of $A$ and prove that it is not the only member of $A$.

Homework Statement Prove or disprove the following (i) ##\forall a\in\mathbb{R}[(\forall \epsilon>0,a<\epsilon)\Leftrightarrow a\leq 0]## 2. The attempt at a solution Can't we disprove the above statement by telling ##a\leq 0 \nRightarrow (\forall \epsilon>0,a<\epsilon)## through a counter...

Please help me to define correctly, in the language of mathematics, the configuration of sets shown on the picture. Homework Statement I'd like to define the following rules: U is a set with infinite members. L is a list or set of properties. Every property (Ls1, Ls2 ... ) have a value (...

I am considering the following question and I want you to agree (but perhaps you don’t):Rutherford wrote a letter to Bohr, as an answer to a previous letter from Bohr containing one of the first of Bohr’s descriptions of the atomic model, saying that he understood the atom model Bohr advocated...

I'm reading the proof that a cauchy sequence is convergent. Let an be a cauchy sequence and let ε=1. Then ∃N∈ℕ such that for all m, n≥N we have an-am<1. Hence, for all n≥N we have an-aN<1 which implies an<aN+1. Therefore, the set {n∈ℕ: an≤aN+1} is infinite and thus {x∈ℝ : {n∈ℕ: an≤x} is...

Whats the difference between a level set and a level curve, with regards to multi-variable calculus?

I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in...

Homework Statement My teacher has notes online that say: A Simple Construction Technique for WellFounded Orders Any function ƒ : S→N defines a wellfounded order on S by x < y iff ƒ(x) < ƒ(y). Example: Lists are wellfounded by length. Binary trees are wellfounded by depth, by number of nodes...

Homework Statement Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.) f(z) = Log(iz+1) / (z^2+2z+5) Homework EquationsThe Attempt at a Solution Not sure how to even attempt this solutions but I wrote down that...

Homework Statement Homework EquationsThe Attempt at a Solution I have managed to solve it for the finite case, where the masure is less than infinity. But how do I solve it if the ,measure if the measure of E is infinite?

Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...