Proofs Definition and 699 Threads

The following is a repost from 2008 from someone else as there was no solution offered or provided I thought id post one here Homework Statement neither my professor nor my TA could figure this out. so they are offering fat extra credit for the following problem Let n be a positive integer...

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} -...

I've been thinking about the properties of the Dirac delta function recently, and having been trying to prove them. I'm not a pure mathematician but come from a physics background, so the following aren't rigorous to the extent of a full proof, but are they correct enough? First I aim to...

Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions. here is my question: Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists. The part that has me confused is the y to the power of 3, where...

Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero. The hint is to use Stirling's approximation. What is this?

I have been studying history of relativity theory and now it seems to me, that it is wrong to automatically assume that proofs of Lorentz covariance are proofs of Special relativity theory. It seems to me, that there is broader group of theories, that are compatible with Lorentz covariance but...

If three variables x,y and z are related via some condition that can be expressed as $$F(x,y,z)=constant$$ then the partial derivatives of the functions are reciprocal, e.g. $$\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$$ Is the correct way to prove this the following...

Let $a$, $b$, and $c$ be integers, where a $\ne$ 0. Then $$ $$ (i) if $a$ | $b$ and $a$ | $c$, then $a$ | ($b+c$) $$ $$ (ii) if $a$ | $b$ and $a$|$bc$ for all integers $c$; $$ $$ (iii) if $a$ |$b$ and $b$|$c$, then $a$|$c$. **Prove that if $a$|$b$ and $b$|$c$ then $a$|$c$ using a column proof...

I wonder which theorem has the most proofs, or has been proven in the most ways? I know of Loomis' The Pythagorean Proposition which came out decades ago & contains 370 proofs & more, & the proofs are even catalogued into four types (algebraic, geometric, etc). So that makes me think the...

Given a function f(x), a point x0, and a positive number E (epsilon), write the limit then find delta>0 such that for all x 0< |x-x0| < delta -> |f(x)-L| < E f(x) = 3-2x, x0=3, E=.02 Here is my attempt: Lim (3-2x) as x->3 = -3 -.02 < |3-2x - 3| <.02 -.02 < |-2x| < .02 .01 > x > -.01 -2.99 > x-3...

I've been reading Wald's book on General Relativity and in chapter 3 he introduces and uses the so-called Hadamard's Lemma: For any smooth (i.e. C^{\infty}) function F: \mathbb{R}^{n}\rightarrow\mathbb{R} and any a=(a^{1},\ldots,a^{n})\in\mathbb{R}^{n} there exist C^{\infty} functions H_{\mu}...

First of all, apologies as I've asked this question before a while ago, but I never felt the issue got resolved on that thread. Is it valid to prove that \int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx using the fundamental theorem of calculus (FTC)?! That is, would it be valid to do...

First, let me say that I am a senior physics undergrad. I have failed Linear Algebra once before. Otherwise I am a straight A student. I'm also taking Ordinary Differential Equations right now, and I breeze through that class without a care in the world. I'm not sure if I've developed some sort...

Homework Statement Prove the following propositions: 1) ∀x ∈ (0, 1), ∃y ∈ (0, 1), x < y and 2) ∀x, y ∈ R, if x < y, then ∀b ∈ (0, ∞), ∃a ∈ (0, ∞), x + ab < y. Can anyone help me out with either one? I have a few others that I can get but I can't get these two. Mainly because these don't...

1) 3^(2^a) + 1 divides 3^(2^b) -1 2) If d > 2, d ∈ N, then d does not divide both 3^(2^a) + 1 and 3^(2^b) -1 Attempt: Set b = s+a for s ∈ N m = 3^(2^a). Then 3^(2^b) - 1 = 3^[(2^a)(2^s)]-1 = m^(2^s) -1 Thus, m+1 and m-1 divides m^(2^s) -1 by induction. If s = 1, then m^(2^s) -1 = m^2 -...

0. Background First and foremost, this is a proof-reading request. I'm going through Velleman's "How To Prove It" because I found that writing and understanding proofs is a prerequisite to serious study of mathematics that I did not meet. Unfortunately, the book is very light on answers to its...

I know that I have already posted a couple of threads like this one (albeit dealing with different courses), but I have had excellent responses here, and I was hoping I could get a couple more. I know that it is best to speak to academic advisors and professors, but there are only a couple that...

Hello PF people. It's my first post here, but I have been lurking around this forum for awhile now. I'm currently learning differential calculus using a text by Stewart and I want to attain a better comprehension of pure mathematics. My question is: would it be a good idea to get another text...

I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...

Does Ross's book teach and/or use Epsilon-delta proof techniques?

I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...

I will even take book recommendations, though I have read Polya's "how to solve it," and Vellemans similarly titled "How to Prove it." I think I am looking more for how to organize my thoughts, and much of this overlaps with "how to study," which, I am still trying to learn how to do. My...

Homework Statement Hypotheses: not a, b or not c, b→ (a and d), e→(c) Conclusion: not e 2. The attempt at a solution: So far, I have this: 1) not a as premise 2) b or not c as premise 3) b→ (a and d) as premise 4) e→(c) as premise 5) a by Step 1 and Law of Excluded Middle. 6) c is true...

Homework Statement [/B] Solve the recurrence relation (use iteration). an = an-1 + 1 + 2n-1 a0 = 0 Then prove the solution by mathematical induction. Homework EquationsThe Attempt at a Solution a1 = 2 a2 = 5 a3 = 10 a4 = 19 a5 = 36 The solution appears to be an = n + 2n - 1 How are we...

In a proof. Prove that **given**: $$\lim_{x \to a} f(x) = L$$ then $$\lim_{x\to a} |f(x)| = |L|$$ We know that $$|f(x) - L| < \epsilon \space \text{for} \space |x - a| < \delta_1$$ What is the objective then? Do we prove there exists a $\delta_2$ such that $\displaystyle \lim_{x\to a}...

Hi, I'm comfortable using a direct proof to prove ##P → Q## type statements when I have a ##P## that is either always true (e.g ##x=x##) or can be true (e.g. ##x > 3##). But what about when ##P## is definitely false, (e.g. ##x \neq x##), or definitely false in relation to an earlier statement...

Hey guys, I have been interested in formalistic mathematics for a while, about a year now. Every time I read a formalistic book on math (Principles of Mathematical Analysis by Rudin is a great example) I never understand how mathematicians develop the structure they present in the books. And...

Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets). I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space). Homework Statement Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0 Homework Equations Given below: The...

This is in reference to a POTW, http://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-135-october-27th-2014-a-12786.html. The logic behind this problem is simple, the number 2^{2^x} can only have factors of 2. But (n + 1)^3 - 1 contains an odd factor. Great...

How would you prove that adding two vectors in the column space would result in another vector in the column space? I know this is maybe the most basic property of vectors and subspaces, and that the very definition of the column space says it's spanned by vectors in the column space. Is there...

Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...

Hey all! I am having some trouble with a certain problem on my homework. I would like some guidance. I have to prove one side of the equation is equal to the other, as you may know, as this is an algebraic proof. This in itself isn't too hard. The hard part is just this one particular problem. I...

Homework Statement Prove whether the function f(x) = x/(1+x^2) with domain & codomain = reals is one-to-one, onto, or both. Homework EquationsThe Attempt at a Solution I know to show if it's one-to-one I have to show a/(1+a^2) = b/(1+b^2), ultimately that a = b, I don't know how to simplify...

Hello all, I've got one more semester before I earn my physics MS, and I have space for one or two extra courses. I am going into oceanography, and I would like to have a strong foundation in math in order to understand the theory I'll encounter as well as possible. Lots of physical...

PLEASE HELP! i am so lost on this. we're using delta epsilon proofs and i am so confused since it was never properly taught to me in calc 1. find the limit. $\lim_{{(x,y)}\to{(0,0)}}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$

The only proof-based math class I've taken so far was on abstract algebra. Concepts were easy for me to understand, but I was constantly having trouble with some of the proofs. I so frequently get this feeling that the last, tiny trivial step left in my proof is just "right there," and yet I...

To those of you who may grade proofs in which the result is not stated (ie prove or disprove), which of the following do you think is easier to grade, and a better format: A) Proof... Therefore the theorem is false/true. B) The theorem is false/true. To see this, consider the following...

Hello guys, I'm new in this forum, this is my first Thread. I've started reading Robert Geroch's Mathematical Physics recently and I've been having problems with some of the proofs that involve monomorphism. He defines monomorphism the following way (pg 4): let ψ be a morphism between A...

Hey guys, I've a few more questions this time around from my problem set: (Ignore question 2abc, I only need help with the first one) Question: For the first one, in order to prove that a function is one-to-one, f(x1) =/ f(x2) when x1 =/ x2. Thus, the horizontal test applies. So I said...

Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X). I really don't know how to get started on this proof but I tried to start with something like this: ∀ m,n ∈ A,B ⇒ m,n ∈ X ⇒ Is this the right way to start on this proof...

To what extent, if any, is an understanding of mathematical proofs required for a scientist? I can empathize with a need for an understanding of the general machinery of the tools you are using (understanding, for example, how it is the chain rule came about, ie, how it was derived) but, using...

Problem 1 Suppose ab=cd, where a, b, c d \in N. Prove that a^{2}+b^{2}+c^{2}+d^{2} is composite. Attempt ab=cd suggests that a=xy, b=zt, c=xz. d=yt. xyzt=xzyt. So (xy)^{2}+(zt)^{2}+(xz)^{2}+(yt)^{2}=x^{2}(y^{2}+z^{2})+t^{2}(z^{2}+y^{2})=(x^{2}+t^{2})(z^{2}+y^{2}) Therefore this is...

Hi, I've been trying a couple of proofs that my calc teacher gave me, but I'm not sure if I have the right approach or not. 1) Prove that the degree of the depressed polynomial is exactly one less than the degree of the original polynomial. - For this proof, all I can come up is the face...

Hello, I have a problem with algebra and divisibility etc. I have a swedish textbook that really sucks. Not a good solutions section and no separate solutions manual either. Just a lot of proofs to show. At the moment I'm stuck at proofs with divisibility. I have two examples: 1)...

Can someone explain to me what is wrong with the following argument? There are two parts. First of all, K-S, despite passing reference to hidden variables, doesn't really seem to depend on any interesting properties of HV, but instead appears to be an indictment of QM itself by asserting that QM...

I'm trying to do some extra course work to prepare for my final next week but I'm having a lot of trouble with the book problems. They talk about a lot of things we weren't taught. Can someone help me out here? Prove: n\niZ, n= a multiple of gcd(a,b) ⇔ n is a linear combination of a and b This...

Homework Statement (a) Prove that if n is an integer and n2 is a multiple of 3, then n is a multiple of 3. (b) Consider a class of n students. In an exam, the class average is k points. Prove, using contradiction, that at least one student must have received at least k marks in the exam...

Homework Statement So I was able to find a problem that was kind of similar to a homework problem that I am working on. Unfortunately, I'm not quite sure what is going on partially within the problem. In the problem they state that \phi=\phi*, but it does not state why. I was wondering...

Homework Statement So I would know how to prove a statement like \sqrt{2} by contradiction, all you have to do is assume to negation. But what about something like p → q Like if p = (bc mod a != 0), q = (b mod a != 0), how would I prove this, would I negate q or p, or both?

Homework Statement (p \wedge t)\rightarrow (r \vee s),q \rightarrow (u \wedge t), u \rightarrow p, \neg s, q, show that these premises imply the conclusion of rThe Attempt at a Solution The question calls for rules for inference to solve this problem, how would I go about doing that...