Proof Definition and 999 Threads

I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml I'd like to know if the following can be considered as a valid and rigorous proof: Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in...

Homework Statement Let ##E'## be the set of all limit points of a set ##E##. Prove that ##E'## is closed. Prove that ##E## and ##\bar E = E \cup E'## have the same limit points. Do ##E## and ##E'## always have the same limit points? Homework Equations Theorem: (i) ##\bar E## is closed (ii)...

Intriguing and informative story on gravity wave detection. Are gravastars an alternative to black holes? Is it possible the there are NO black holes? The collapse of mass into a ball of energy that presses out and stabilizes the incoming mass is a thought provoking alternative to the common...

Homework Statement . Disprove the following statement: There exists integers a, b, c, none divisible by 7, such that 7|a^3 + b^3 + c^3 Homework EquationsThe Attempt at a Solution if 7|a^3 + b^3 + c^3, then a^3 + b^3 + c^3 is congruent to 0(mod 7) if a,b,c are none divisible by 7 then I just...

Homework Statement Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n. Homework EquationsThe Attempt at a Solution I tried different approaches such as...

Homework Statement Reading Feynman The Principle of Least Action out of The Feynman Lectures on Physics, Vol 2. Link to text http://www.feynmanlectures.caltech.edu/II_19.html So I'm having a problem proving that, section 19-2 5th paragraf, that "Now the mean square of something that deviates...

Homework Statement Prove that 3^n>n^4 for all n in N , n>=8 Homework Equations The Attempt at a Solution Base case: 3^8>8^4 Inductive step Assume 3^n>n^4. Show 3^n+1>(n+1)^4 I tried a lot of approaches to get from the inductive hypothesis to what I want to show Ex: 3^n>n^4 3^n+1>3n^4...

Prove that the number of unordered partitions of an even number 2n into 2 composites is greater than the number of unordered partitions of an odd number 2n+1 into 2 composites for n>1 and n\ne p prime.

Homework Statement Homework Equations let the point where the line through B and X intersects with AC be P The Attempt at a Solution [/B] I know that ACdotBP = 0 AC = AD+DC BP = PC+ CD Therefore (AD+DC)dot(PC+CD)=0 I also know that: ECdotCE = 0 BCdotDA=0 However I am stuck on...

Homework Statement Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd. Homework EquationsThe Attempt at a Solution It's a little bit intuitive. Proof 1 and 3 have the same parity. They are both odd so if(odd)=odd then...

Homework Statement Proof 2/5*(2^0.5)-1/7 is irrational Homework EquationsThe Attempt at a Solution I did this by splitting the expression and setting contradictions 2/5->rational 2^0.5->irrational Proof first rational times irrational is irrational Proof by contradiction Assume the product...

Homework Statement I am trying to understand the optimality proof of the earliest finish time algorithm. I have attached the pdf which I am reading. It's just 2 pages. I don't understand what they mean with solution still feasible and optimal (but contradicts maximality of r). An explanation...

Homework Statement (a) If e is part of some MST of G, then it must be a lightest edge in some cutset of G. Homework Equations Cut property The Attempt at a Solution When the cutset has just one edge then yes it's true obviously. I am think I can do this by contradiction. Assuming e_i is part...

Mod note: Member warned that homework questions must be posted in the Homework & Coursework sections http://imgur.com/zGB2dnY Was given this problem a few weeks ago and I'm not sure how I should be approaching it. Please let me know which theorems I should look into in order to solve the problem.

I have attached two images from my textbook one of which is a diagram and the other a paragraph with which I am having problems. The last sentence mentions that due to violation of 2nd law we cannot convert all the heat to work in this thermodynamic cycle. However what is preventing the carnot...

In de Broglie's original proof of the theorem of phase harmony, the frequency of the moving wave of energy mc^2 (not the internal periodic phenomenon wave) is multiplied by the following term ##freq * ( t - \frac{\beta * x}{c} ) ## Does anyone have an idea where the fraction comes from? All...

Homework Statement Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order. Homework Equations Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html The Attempt at a Solution Since y is a conjugate of x...

I would like to prove the Lorentz invariance of the Klein-Gordon equation by proving the invariance of the action ##\mathcal{S} = \int d^{4}x\ \mathcal{L}_{KG}## under a Lorentz tranformation. I would like to do this by first proving the Lorentz invariance of the ##\mathcal{L}_{KG}## and then...

Homework Statement Let a,b be in the positive reals. Prove a/b+b/a is >=2 Homework EquationsThe Attempt at a Solution I have no idea. Maybe add the two ratios: (a^2+b^2)/a*b and then try to analyze separately the numerator and denominator?

This question will essentially be more of a how-to plea or general help request. I'm currently studying math and I'm at the point where I've transitioned into upper-division classes, most if not all of which are proof based. To be blunt, I currently feel discouraged at the prospect of being...

Homework Statement Assume {B, C, D} is a partition of the universal set U, A is a subset of U and A is not a subset of B complement, A is not a subset of C complement, A is not a subset of D complement. Prove that {A ∩ B, A ∩ C, A ∩ D} is a partition of A. Homework EquationsThe Attempt at a...

Homework Statement Given an array of positive integers A[1, . . . , n], and an integer M > 0, you want to partition the array into segments A[1, . . . , i1], A[i_1 + 1, . . . , i2], . . . , A[i_k−1 + 1, . . . , ik], so that the sum of integers in every segment does not exceed M, while...

Homework Statement Given a function g(t)=acosωt + bsinωt, where a and b are constants, show that g(t) is the real part of the complex function: keiΦeiωt for some k and Φ Remark: the complex expression keiΦ is called a phasor. If we know that g(t) has the form kcos(ωt+Φ) then we need know only...

Homework Statement Suppose the matrix A with real entries has the complex eigenvalue λ=α+iβ, β does not equal 0. Let Y0 be an eigenvector for λ and write Y0=Y1 +iY2 , where Y1 =(x1, y1) and Y2 =(x2, y2) have real entries. Show that Y1 and Y2 are linearly independent. [Hint: Suppose they are...

Homework Statement Let ##f:S\to T## be a given function. Show the following statements are equivalent: a) ##f## is 1-1 b) ##f(A\cap B) = f(A) \cap f(B),\; \forall A,B \in S## c) ##f^{-1}(f(A)) = A,\; \forall A \subseteq S.## Homework Equations Definition: ##f## is 1-1 of ##A## into ##B##...

Homework Statement Suppose m, n are relatively prime. In the problem you will prove the key property of Euler’s function that φ(mn) = φ(m)φ(n). (a) Prove that for any a, b, there is an x such that x ##\equiv## a (mod m), (1) x ##\equiv## b (mod n). (2) Hint: Congruence (1) holds iff x...

Homework Statement Show that the set ##\{x \in \mathbf Q; x^2< 2 \}## has no least upper bound in ##\mathbf Q##; using that if ##r## were one then ##r^2=2##. Do this assuming that the real field haven't been constructed. Homework Equations N/A The Attempt at a Solution Attempt at proof: ##r\in...

Homework Statement Prove that there does not exist a continuous function f, defined on R which takes on every value exactly twice. Homework Equations It uses this property: 1... If f is continuous on [a,b], then there exists some y in [a,b], such that f(y)≥f(x), for all x in [a,b]The Attempt...

Homework Statement Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x \in A##. Prove that ##\inf A = -\sup(-A)##. Homework Equations Definition: Suppose ##S## is an ordered set, ##E\subset S##, and ##E## is bounded above...

I do not get some points of this proof about the time derivative of a unit vector $\hat{u}$ (costant magnitude) which is following a precession motion. The picture is the following. I want to prove that $$\frac{d\hat{u}}{dt}=\vec{\Omega}\wedge \hat{u}.$$ I'm ok with almost all the proof except...

Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T. Attempt: I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b i know that a> s and b< t for all s and t. How do i continue? , do i prove it...

Homework Statement Hi everybody! I'm having a hard time to find a way to cleanly prove that ∫(1/ln(x)) dx between 1 and 2 doesn't exist. At first I thought it was because it's not bounded (Riemann criterion I believe), but then I looked at another unbounded definite integral such as ∫lnx dx...

Homework Statement Consider the following sorting algorithm for an array of numbers (Assume the size n of the array is divisible by 3): • Sort the initial 2/3 of the array. • Sort the final 2/3 and then again the initial 2/3. Reason that this algorithm properly sorts the array. What is its...

Does there exist a proof of the following: It is well known that Picard successive approximations on the Fredholm-equation (1) $y(x)=f(x)+{\lambda}_{1}\int_{a}^{b} \,k(x,s)y(s)ds$ written in operator form as $y=f+{\lambda}_{1} Ky$ converges if (2) $|{\lambda}_{1}|. ||K||<1$ where $K$...

Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...

Hello, I am currently preparing myself for exams and I have a past exam question which I can't solve. This question concerns online learning and the following picture illustrates it: Is anyone able to help me out and propose a solution to this question?

Homework Statement In a binary tree all nodes are either internal or they are leaves. In our definition, internal nodes always have two children and leaves have zero children. Prove that for such trees, the number of leaves is always one more than the number of internal nodes. Homework...

Is it always true that if I have any system of resistors and I calculate the resistance between two points, when I decrease the resistance of one resistor, then the resistance measured between the same two points as previously will not increase? I.E if i have lots of resistors between two...

Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution The trick to solve this problem is that when we assume that it is true for a 2^n x 2^n matrix and then we expand this matrix with 1's to a 2^n+1 x 2^n+1, we can divide the resulting matrix into 4 submatrices...

Homework Statement If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element. Homework EquationsThe Attempt at a Solution I feel like there is missing information, but that cannot be. This seems too simple: The order of G is the smallest possible integer n...

Homework Statement . Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A. Homework EquationsThe Attempt at a Solution I know...

Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution So I have to show that each side is a subset of the other side Assume x∈ A ∪ (∩Bi) so x∈A or x∈∩Bi case 1 x∈ ∩ Bi so x∈ (B1∩B2∩B3...∩Bn) which implies x∈B1 and x∈B2 ... and x∈Bn so x∈B1∪A and x∈B2∪A...

Of course 1 isn't same as -1. This proof must be wrong but I can't find which part of this proof is wrong. Could you help me with this problem? (1)$$1 = \sqrt{1}$$ (2)$$= \sqrt{(-1)(-1)}$$ (3)$$= \sqrt{(-1)} \cdot i$$ (4)$$= i \cdot i$$ $$=-1$$

Homework Statement I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}## ##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one Homework EquationsThe Attempt at a Solution I know for certain that this function is not onto given the codomain of real...

Homework Statement http://puu.sh/nYQqE/2b0eaf2720.png Homework Equations http://puu.sh/nYSjQ/e48cad3a8b.png The Attempt at a Solution http://puu.sh/nYYjW/174ad8267c.png My main issue is with part b) and part d). I think that part b) is mostly right, but part d) is definitely wrong and...

I am trying to prove that two definitions of a finite set are equivalent. 1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## ) 2.) A set ##A## is finite if and only if...

I'm trying to prove that kinetic energy is not conserved in inelastic collisions using the conservation of momentum. This is the set-up. An object A of momentum ##{m_1}{v_1}## collides inelastically with object B of momentum ##{m_2}{v_2}## using momentum conservation ##P_i = P_f## {m_1}{v_1} +...

I'm preparing for college on my own. I need to proof that: [p -> (q v r)] and [(p ^ -q) -> r] are logically equivalent. with 1) v "or" 2) ^ "and" 3) -q "negation of q" I did this using truth tables and this perfectly shows that those 2 statements are logically equivalent. Can someone confirm...

Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?

In the process of doing a proof by induction, can you use a contradiction to show that if P(k) holds then P(k+1) must hold? What I mean is, after establishing that P(0) holds, can I assume that P(k) holds and that P(k+1) does not, and show that a contradiction arises, and thus conclude that if...