Proof Definition and 999 Threads

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

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  1. Mr Davis 97

    Linear Independence of a Set of Vectors

    Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...
  2. binbagsss

    Q about the proof of periods of non-constant meromorphic functions

    Homework Statement [/B] Theorem attached. I know the theorem holds for a discrete subgroup of ##C## more generally, ##C## the complex plane, and that the set of periods of a non-constant meromorphic function are a discrete subset. I have a question on part of the proof (showing the second...
  3. A

    Vector Proof Homework: The Rotation Matrix

    Homework Statement Homework Equations The Rotation Matrix The Attempt at a Solution I am sorry but I do not know how to even begin
  4. X

    (Number theory) Sum of three squares solution proof

    Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...
  5. L

    Proof Check: Geometry AB=EF If A=/B

    Homework Statement Let A and B be elements of the line EF such that A=/B prove that the line AB=EFHomework Equations Axiom that two points determine a unique line and that the intersection of two lines has two distinct points then these lines are the same. The Attempt at a Solution [/B] If A...
  6. binbagsss

    I Proving Hurwitz Identity: Modular Forms & Beyond

    Hi, My notes say that hurwitz identity currently has no elementary proof? One way to prove the identity is through modular forms: to consider Eisenstein series, ##E_4^2## and ##E_8## , note that the dimension of space of modular functions of weight 8 is one, find the constant of...
  7. binbagsss

    Q about a Proof -- periods meromorphic function form discrete set

    Homework Statement Hi, As part of the proof that : the set of periods ##\Omega_f ## of periods of a meromorphic ##f: U \to \hat{C} ##, ##U## an open set and ##\hat{C}=C \cup \infty ##, ##C## the complex plane, form a discrete set of ##C## when ##f## is a non-constant a step taken in the...
  8. binbagsss

    Sin inequality proof , ##0 \leq 2x/\pi \leq sin x##

    Homework Statement Homework EquationsThe Attempt at a Solution Hi How do I go about showing ##0 \leq \frac{2x}{\pi} \leq sin x ##? for ## 0 \leq x \leq \pi /2 ## I am completely stuck where to start. Many thanks. (I see it is a step in the proof of Jordan's lemma, but I'm not interested in...
  9. T

    Using symbolic logic in mathematical proof?

    is this a practical way of proving math theorems? i asked because when i tried, it seemed difficult for me to decide as to how exactly i should translate theorems and given statements into logical forms and since there are so many different ways, i do not know which one is correct. For example...
  10. VrhoZna

    Proof regarding direct sum of the dual space of a v-space

    (From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V. 1. Homework Statement Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let $$V_j = W_1 + \cdots + W_{j-1}...
  11. S

    I Why isn't tunnelling considered proof of hidden variables?

    When I hear that mass of a particle has managed to hop through a solid barrier ..it tells me that the mass was a variable and not physical at the time.
  12. L

    I How can I improve my proof skills with internet resources?

    Is this proof even correct?! It places assumption on a and c NOT BEING ZERO. Thanks in advance. I am new to proofs.
  13. evinda

    MHB Questions about proof of theorem

    Hello! (Wave) We say that the space $\Omega$ satisfies the exterior sphere condition at the point $x_0 \in \partial{\Omega}$ if there is a $y \notin \overline{\Omega}$ and a number $R>0$ such that $\overline{\Omega} \cap \overline{B_y(R)}=\{ x_0 \}$. Let the function $\phi \in...
  14. B

    Verifying a Proof about Maximal Subgroups of Cyclic Groups

    Homework Statement Show that if ##G = \langle x \rangle## is a cyclic group of order ##n \ge 1##, then a subgroup ##H## is maximal; if and only if ##H = \langle x^p \rangle## for some prime ##p## dividing ##n## Homework Equations A subgroup ##H## is called maximal if ##H \neq G## and the only...
  15. I

    How Do You Prove a Function is Not Uniformly Continuous?

    Homework Statement Let ##f:X \to Y##. Show that ##f## not uniform continuous on ##X## ##\Longleftrightarrow## ##\exists \epsilon > 0## and sequences ##(p_n), (q_n)## in ##X## so that ##d_X(p_n,q_n)\to 0 ## while ##d_Y(f(p_n),f(q_n))\ge \epsilon##. Homework Equations Let ##f:X\to Y##. We say...
  16. I

    Is a Complete Subspace Necessarily Closed in a Metric Space?

    Homework Statement Let ##E## be a metric subspace to ##M##. Show that ##E## is closed in ##M## if ##E## is complete. Show the converse if ##M## is complete. Homework Equations A set ##E## is closed if every limit point is part of ##E##. We denote the set of all limit points ##E'##. A point...
  17. I

    Convergence of sequence in metric space proof

    Homework Statement Let ##E \subseteq M##, where ##M## is a metric space. Show that ##p\in \overline E = E\cup E' \Longleftrightarrow## there exists a sequence ##(p_n)## in ##E## that converges to ##p##. ##E'## is the set of limit points to ##E## and hence ##\overline E## is the closure of...
  18. CynicusRex

    Prove that if a² + ab + b² = 0 then a = 0 and b = 0

    Homework Statement Prove that if a² + ab + b² = 0 then a = 0 and b = 0 Hint: Recall the factorization of a³-b³. (Another solution will be discussed later when speaking about quadratic equations.) Homework Equations a² + ab + b² is close to a² + 2ab + b² = (a+b)² a³-b³=(a-b)(a²+ab+b²) The...
  19. mastrofoffi

    Proof of Dirichlet stability theorem

    Hola, I tried to give a proof of this theorem and then check it against the one given by my book(Fasano, Marmi - Analytical Mechanics); I feel like mine seems reasonable and pretty intuitive, but the one on the book is a bit different and I don't really understand it completely, so I'd like to...
  20. S

    MHB Is the Russell's Paradox Resolved in Predicate Calculus?

    Prove (formall y) in predicate calculus : $\neg\exists y\,\forall x\,(x\in y\leftrightarrow \neg x\in x)$.
  21. binbagsss

    Delta property, integration by parts, heaviside simple property proof

    Homework Statement I am trying to show that ## \int \delta (x-a) \delta (x-c) dx = \delta (-a-c) ## via integeration by parts, but instead I am getting ##\delta (c-a) ## (or ##\delta (a-c)## depending how I go...). Can someone please help me out where I've gone wrong: struggling to spot it...
  22. D

    Proof f(x)>g(x) in an interval

    Homework Statement let fx, gx be continuous in [a,b] and differentiable in (a,b). at the end of the interval f(a) >= g(a). and f'(x) >g'(x) for a<x<b. proof f(x) > g(x) for a<x<=b Attempt: There is a statement says that if the f'x = g'x for x in [a,b] , then there exists k such that f'x -...
  23. D

    Proof that the line intersects the curve 3 times exactly

    Homework Statement p(x) = 0.2*(x-1)^5, q(x) = 4x-6 The Attempt at a Solution I took the diffrence h(x) = p(x) - q(x) h'(x) = ((x-1)^4) - 4 got two solutions for h'(x)=0.
  24. S

    Implicit function theorem proof question

    Homework Statement I understand the proof of the implicit function theorem up to the point in which I have included a photo. This portion serves to prove the familiar equation for the implicit solution f(x,y) of F(x,y,z)=c. My confusion arises between equations 8.1-4 and 8.1-5 when it is stated...
  25. muimerp

    On the proof of the Principle of Virtual Work

    I'm having some trouble understanding the proof of the Principle of Virtual Work for deformable bodies. I'll give below the proof that I've read, and, next, I'll remark what I'm not understanding. The first thing to remember before going through the proof is that the virtual work done by a...
  26. F

    A Is this a correct proof of the Riemann Hypothesis?

    Found an article online detailing a proof of the Riemann Hypothesis: << link deleted by mentor - unacceptable source >>
  27. J

    What is the proof for n(n^4 - 1) = 10Q for some values Q and n being integers?

    For some values Q and n being integers, prove that n(n^4 - 1) = 10Q. So I've tried this with induction, but it gets pretty messy pretty quickly. So I can see that the LHS will be even no matter what, but I'm not sure where to go beyond this.
  28. G

    Checking a proof of a basic property of prime numbers

    Homework Statement Prove: If p is prime and m, n are positive integers such that p divides mn, then either p divides n or p divides m. Is anyone willing to look through this proof and give me comments on the following: a) my reasoning within the strategy I chose (validity, any constraints or...
  29. V

    A Proof of quantum correlation functions

    Reading through David Tong lecture notes on QFT.On pages 76, he gives a proof on correlation functions . See below link: [QFT notes by Tong][1] [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfI am following the proof steps to obtain equation (3.95). But several intermediate steps of the...
  30. J

    A Need a Proof that Action-Angle Coordinates are Periodic

    Can someone point me to a proof that Action-Angle coordinates in Hamilton-Jacobi Theory must be periodic. I have looked all over and no one seems to prove it, they just assume it. Thanks.
  31. Dtriction

    I Can this method be used to prove the Collatz Conjecture?

    There is a graph showing n on its x-axis and its total stopping time on its y axis. From here we can see that the points on the graph are not random at all; they have some kind of geometric pattern that is due to the 3x+1 in the odd case and x/2 in the even case. I have seen many attempts to...
  32. A

    Proof of this limit formula for e

    Homework Statement http://prntscr.com/dcfe0u Homework EquationsThe Attempt at a Solution So I'm not really strong in proofs but I think you may be able to do something like this:$$lnL = \frac{ln(1+1/x)}{x}$$ $$lnL = \frac{1/x^2}{1+1/x}$$ and then more simplifying I get something like: $$lnL =...
  33. M

    Is this a valid inductive proof?

    Mentor note: moved to the homework section Claim: all numbers divisible by 4 are divisible by 2. Premise: let p(n) return 4n, I.e., the function covers all numbers divisible by four. Reasoning: Let n equal 1 in the base case P(n) is divisible by 2 P(n+1) is divisible by 2 By induction all...
  34. Z

    I Proof check: S in C Compact implies S is closed and bounded

    I am using Lang's book on complex analysis, i am trying to reprove theorem 4.1 which is a simple theorem: Let Compact(S \in \mathbb{C}) \iff Closed(S) \land Bounded(S) I will show my attempt on one direction of the proof only, before even trying the other direction. Assume S is compact Idea...
  35. H

    I Validity of proof of Cauchy-Schwarz inequality

    Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero. If neither x nor y is zero, then by x · y = | x | | y | cos θ, |x · y|=| x | | y | cos θ | ≤ | x | | y | since -1 ≤ cos θ ≤ 1 How valid is this a proof of the...
  36. J

    B Simple proof of Bell's theorem

    The thread I wanted to post my question on got closed. Recapitulating: The best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester): Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When...
  37. G

    I Is Zorn's Lemma Proven? A Closer Look at the Proof

    Salutations, friends from afar. The question I have is mundane, but I felt I should be sure. It is basically to spot the insufficiency in this proof for Zorn's Lemma: If every chain in a partially ordered set M has an upper bound, then M contains a maximal element. Proof: 1. For a set X, take...
  38. H

    Proof that entropy can never decrease

    What is the proof of the fact that for an isolated system, entropy can never decrease?
  39. K

    Math proof: Linear Independence

    Homework Statement How can I show that if a vector (in a vector space V) cannot be written as a linear combination of a linearly independent set of vectors (also in space V) then that vector is linearly independent to the set? Homework Equations To really prove this rigorously it would make...
  40. Nipuna Weerasekara

    How Does the Condition x > -1 Influence the Inequality x² + 1/(x²+1) ≥ 1?

    Homework Statement Let ##x\in \mathbb{R} ## Prove the conditional statement that, if ## x>-1## then ## x^2 + \frac {1}{x^2+1} \geq 1## 2. The attempt at a solution Suppose ## x>-1## is true. Then ## x^2>1## Then ## \frac{1}{2}>\frac {1}{x^2+1}## Then ##x^2+ \frac{1}{2}>x^2+\frac...
  41. Jaroslav

    Prove divisibility, mathematical induction

    I'm still learning English, had to use dictionary and translator, so I'm sorry if its unclear, i will try to explain it more if needed. Homework Statement For n belonging to N when n is even and n > 3, prove that (4^(n-3) + 5^(n-3) + 9) is divisible by 9 Homework Equations 3. The Attempt at...
  42. A

    Mathematical proof for oil drilling

    hi guys, I'm supposed to write a paper and do some research on all aspects of drilling regarding torque and force need to drill an oil well and everything from start to finish and to provide mathematical proof and calculations. I don't know where to start and what are the equations used to...
  43. B

    B Flaw in my proof of something impossible

    Given :- $$g(f(x_1)) = g(f(x_2)) \implies x_1 = x_2$$ Question :- Check whether ##g(x)## is injective or not. Now this is of-course false; counter examples are easy to provide. But I proved that ##g(x)## must be one-one even after knowing the fact it must not. Here is the proof :- Let...
  44. P

    B What's wrong with this proof of sin(i)=0?

    We have, e^(ix)=cosx+isinx So, e^(i*i)=cosi+isini Or e^-1=cosi+isini Or 1/e + 0*i= cosi+isini So, cosi=1/e and sini=0 But that's not the value of sin(i) that I found on the internet. These values are not even satisfying cos^2(x)+sin^2(x)=1. What did I miss?
  45. A

    Torque required to tighten the cap for leak proof

    Hello Anyone, Could you help me in finding the torque req. for a cap to leak proof? My cap (polyproplene) which dia. was 32mm and its detail specs are, thread major dia.- 28.5mm, min. dia. - 26mm, pitch - 3mm, thread angle-45deg which has a EPDM rubber seal placed inside (outer dia 26.5mm &...
  46. bananabandana

    General proof of Arc Length For Parametrised Coodrdinates

    Homework Statement Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length: $$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$...
  47. D

    A Proof of expansion of a certain value

    How do I begin proving: sum(k>=1)8/(k^4+4)=pi*coth(pi)-1? I got this from Mathematica. Thanks in advance for any help.
  48. PWiz

    A proof in the Hilbert-style axiom system

    Homework Statement Provide a complete formal proof that ## \vdash ((A \rightarrow B) \rightarrow C) \rightarrow (B \rightarrow C)##. Homework Equations I am only allowed to use modus ponens and these four 'sentential logic' axioms: A1 ## \neg \alpha \rightarrow (\alpha \rightarrow \beta)## A2...
  49. DoobleD

    B Proof / derivation of d'Alembert principle?

    I can't find a derivation of d'Alembert principle. Wikipédia says there is no general proof of it. Same with stackexchange. I find it surprising so I thought I'd come here to check with you guys. D'Alembert principle has indeed no proof ?
  50. N

    Proving Natural Log Proof: ln|1+σx|

    Homework Statement Prove the following statement: ln|1+\sigma x | = \frac{1}{2} ln|1-x^2| + \frac{\sigma}{2} ln| \frac{ |1+x|}{|1-x|} Homework EquationsThe Attempt at a Solution Starting from right to left would be easier: = \frac{1}{2} ln|(1+x)(1-x)| + \frac{\sigma}{2} ln| 1+x| -...
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