What is Proof: Definition and 999 Discussions

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. K

    A Matrix proof of Euler's theorem of rotation

    The question arises the way Goldstein proves Euler theorem (3rd Ed pg 150-156 ) which says: " In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point"...
  2. Andreas S-H

    B Proving Special Relativity w/ Standard Equipment

    Hello everyone, I am preparing to write an exam project in college about special relativity, however i am missing the critical experiment to prove that it is true. I thought about using the life time of muons, but i don't have a scintillator to detect them, so unless a standard geiger counter...
  3. B

    MHB Proof of Complex Numbers: Delta*w(z, z) Explained

    Hi, I have this problem and its solution but i know what right size is, but i don't understand what left size (delta*w(z, z)) is equal to
  4. M

    Can anyone please check/verify this proof about rational numbers?

    Show sqrt(3), sqrt(5), sqrt(7), sqrt(24), and sqrt(31) are not rational numbers.
  5. M

    Can anyone please check/confirm my work on this proof?

    Please check/confirm my work of this proof and tell me if it's correct or not. Thank you.
  6. S

    I I think I've hit upon a very easy proof of impossibility of quintic

    The discriminant for a monic polynomial having only the free term { a0 } (and the monic degree term of course) is: Δn:0 = σ0 nn a0( n - 1 ) where n is the degree of the polynomial, and σk is a sign (i.e., +1 or -1) while the discriminant for a monic polynomial having only the x1 term (and...
  7. docnet

    A proof that a union of compact spaces is compact

    Prove that if ##X## is a topological space, and ##S_i \subset X## is a finite collection of compact subspaces, then their union ##S_1 \cup \cdots \cup S_n## is also compact. ##S_i \subset X## is compact ##\therefore \forall S_i, \exists## a finite open cover ##\mathcal J_i=\{U_j\}_{j\in...
  8. strangerep

    A Proof that neutrino flavor oscillation implies nonzero neutrino mass?

    [This is a reference request.] I'm dissatisfied with the "proofs" I've found so far. E.g., in Kayser's review from 2008, in the paragraph following his eq(1.4), he assumes a propagation amplitude Prop##(\nu_i)## of ##\exp(-im_i \tau_i)##, where "##m_i## is the mass of the ##\nu_i## and...
  9. S

    I What evidence supports the theory of gluon masslessness?

    Gluons are supposed to have precisely 0 rest mass. However, gluons are always colour confined into hadrons with binding energies of hundreds of MeV. How is gluons´ lack of rest mass proven? Presumably through some symmetries, or lack of some processes. Which kinds of asymmetries and processes...
  10. K

    A Dissipation function is homogeneous in ##\dot{q}## second degree proof

    We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...
  11. C

    Simple Induction Induction proof of Polynomial Division Theorem

    Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...
  12. L

    Analysis 1 Homework Help with Complex Numbers

    I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in. Bar(a+bi/c+di)= (a-bi) / (c-di) Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di)) Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...
  13. Eclair_de_XII

    B Is it invalid to redefine the sgn function in this way in a proof?

    Let ##S_n## denote ##\{1,\ldots,n\}##, where ##n\in\mathbb{N}##. Recall that the ##\textrm{sgn}## function maps a permutation of ##S_n## to an element in ##\{1,0,-1\}##. We want to rework the definition of ##\textrm{sgn}## because it is not sufficient for some proofs about determinants. For...
  14. LukasMont

    Proving A Must Be of Rank 2: The 2x2 Matrix Dilemma

    My trouble is being to show A must be of rank 2. Any ideas?
  15. Lynch101

    B Proof Pertaining to Possible Paths Through 3 Dimensions

    I'll try to phrase this as clearly as possible but my use of terminology might need to be refined. That may be what ultimately comes of this thread, but hopefully the question as I phrase it will make enough sense. I'm not necessarily asking that a proof be provided, rather, I am interested to...
  16. vibha_ganji

    I Proof of 1/(x^2) Not Having Limit at 0

    In Apostol’s Calculus (Pg. 130) they are proving that 1/(x^2) does not have a limit at 0. In the proof, I am unable to understand how they conclude from the fact that the value of f(x) when 0 < x < 1/(A+2) is greater than (A+2)^2 which is greater than A+2 that every neighborhood N(0) contains...
  17. steve1763

    A Quantum linear code/ Dual Code (CSS) proof

  18. C

    DP: proving existence of optimal substructure for "Sherlock and Cost"

    I was attempting to solve the "Sherlock and Cost" problem from HackerRank using DP: But before I went to come up with a recursive relation, I wanted to find if the problem possesses an optimal substructure, and I was following these steps as written at CLRS book: Mentor note: Inline images of...
  19. M

    Proof by contradiction - any non-zero number divided by itself is 1

    Proof by contradiction (for some reason the LaTeX code is not working for me. Sorry) Lets assume that A, B, and C are non-zero real numbers; A = B ; and C is not equal to 1. A/ B = C A = B x C But BxC could be equal to B, if and only if C =1 Also, could you recommend a book where I...
  20. K

    I What's the proof that R2 /(R+h) = (1 – 2h/R)

    Hi, On this link: https://physicsteacher.in/2020/07/11/the-formula-for-acceleration-due-to-gravity-at-height-h-with-derivation/ They prove the formula for acceleration due to gravity at height h, which is: g1 = g (1 – 2h/R). There are similar articles online. When they go through the last...
  21. Delta2

    I From a proof on directional derivatives

    Given that the partial derivatives of a function ##f(x,y)## exist and are continuous, how can we prove that the following limit $$\lim_{h\to 0}\frac{f(x+hv_x,y+hv_y)-f(x,y+hv_y)}{h}=v_x\frac{\partial f}{\partial x}(x,y)$$ I can understand why the factor ##v_x## (which is viewed as a constant )...
  22. WMDhamnekar

    MHB Proof of known velocity of wind

    My attempt to answer this question: Let the actual velocity of wind is $\vec{v}=x\hat{i} + y\hat{j}$ where $\hat{i}$ and $\hat{j} $ represents velocities of 1KM per hour towards east and north respectively. As the person is going northeast with a velocity of 6KM/hr, his actual velocity is $...
  23. J

    Proof involving exponential of anticommuting operators

    For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that $$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$ which generalizes what I initially...
  24. A

    Linear Algebra - LU Factorization

    Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance. Problem: Work:
  25. George Keeling

    I Uncovering a Decent Proof of E=mc^2

    Sean Carroll says that in SR the time component of the 4-momentum of a particle is its energy. It is of course also ##mc^2dt/d\tau##. He uses that to prove that ##E=mc^2##. Which begs the question why does ##E=p^0##? Misner, Thorne, Wheeler do roughly the same thing. I find these 'proofs'...
  26. W

    I Is there a "Nice" proof that R^2 is not disconnected when we remove 2 points

    Are there "nice" ( without heavy machinery) proofs that ## X:=R^2 - \{p,q\} ## is connected? All I can think is using that path-connectedness implies connectedness. So we consider x,y in X and show there is a path joining them. I am looking for an argument at undergrad level, so that I would not...
  27. Lecture 5a - Pandemic Pedantics - Derivation of PCA and Kernel PCA

    Lecture 5a - Pandemic Pedantics - Derivation of PCA and Kernel PCA

    Here we talk about how we come to the formulas for PCA and Kernel PCA. We briefly introduce kernel functions, and talk about feature spaces. This builds on the introductory lecture for PCA and also that for Kernel PCA.
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  28. B

    Multivariable Calculus proof for Optics

    Part A) For part A I forgo breaking down the identity into it's component x, y, and z parts, and just take the r derivative treating r' as a constant vector. This seems to give the right answer, but to be entirely honest I'm not sure how I'd go about doing this component by component. I figure...
  29. T

    A Lie Bracket * Matrix * vector (Need proof)

    As an aside, fresh_42 commented and I made an error in my post that is now fixed. His comment, below, is not valid (my fault), in that THIS post is now fixed.Assume s and w are components of vectors, both in the same frame Assume S and W are skew symmetric matrices formed from the vector...
  30. malawi_glenn

    Help with a proof regarding convergent sequence (proof by contradiction)

    Ok I am trying to brush up my real analysis skills so that I can study some topology and measure theory at some point. I found this theorem in my notes, that is proven by using proof by contradiction. However, I have a hard time understanding what the contradiction really is... Here is the...
  31. A

    MHB Proof that lim loga_n/n = 0 in epsilon delta language

    Let $\,a>0\,,\,a\neq1\,$ be a real number. We can prove by using the continuity of $\ln n$ function that $\;\lim\limits_{n\to\infty}\dfrac{\log_an}n=0\;$ However, this problem appears in my problems book quite early right after the definition of $\epsilon$-language definition of limit of a...
  32. N

    Proof Involving the Diagonals of Parallelogram

    Chapter 1, Section 1.1. Look at the picture. Question 57. Let me see. To show this prove, I must find the midpoint of the diagonals. The midpoint of (b, c) and (a, 0) must be the same as the midpoint of (0, 0) and (a + b, c). You say?
  33. Twigg

    A Proof of Classical Fluctuation-Dissipation Theorem

    Sorry if there's latex errors. My internet connection is so bad I can't preview. Here's the wikipedia proof I'm referring to. I'm fine with the steps up to $$W(x,0) = W_0 (x) [1 + \beta f_0 (x(0) - \langle x \rangle_0) ]$$ where ##W(x,t)## is the probability density of finding the system at...
  34. L

    Statistics: Verifying a Probability Proof

  35. HrvojeDjurdjevic

    I Is the proof of the KRK endgame theorem rigorous and original?

    The articles: https://lmcs.episciences.org/5328/pdf http://argo.matf.bg.ac.rs/publications/2013/2013-icga-krk-sat.pdf http://archive.ceciis.foi.hr/app/public/conferences/1/papers2012/dkb3.pdf KRK endgame is a win for white regardless of starting position, with the trivial drawing exception in...
  36. A

    I How can the Lp Norm be used to prove inequalities?

  37. T

    Experimental proof of Venturi Effect

    Venturi effect is known for centuries. And most probably that's why experimental proofs are rare because it's already accepted. But, I want to know how close real results are in case of experiments regarding Venturi Effect. I am especially interested in results of experiment regarding velocity...
  38. A

    I Proving the Finite Binomial Series for k Non-Negative Integer

    Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!
  39. Samkiwi

    How is the Relativity Formula Proven for an Electron in an Electric Field?

    It is an electron initially pushed by the action of the electric field. The vectors of force and velocity are parallel to each other. Here's the questionA possible expression of speed as a function of time is the following: $$v(t) = \frac{At}{\sqrt{1 + (\frac{At}{c})^2}}$$where is it $$A...
  40. Samkiwi

    Deriving and Verifying the Relativity Formula for Electron Speed

    It is an electron initially pushed by the action of the electric field. The vectors of force and velocity are parallel to each other. Here's the questionA possible expression of speed as a function of time is the following: $$v(t) = \frac{At}{\sqrt{1 + (\frac{At}{c})^2}}$$where is it $$A...
  41. Eclair_de_XII

    B Is this a valid proof for the Extreme Value Theorem?

    If ##f## is a constant function, then choose any point ##x_0##. For any ##x\in K##, ##f(x_0)\geq f(x)## and there is a point ##x_0\in K## s.t. ##f(x_0)=\sup f(K)=\sup\{f(x_0)\}=f(x_0)##. Now assume that ##f## is not a constant function. Construct a sequence of points ##x_n\in K## as follows...
  42. Z

    Proving Energy Conservation in a Gravitational System with Multiple Bodies

    Hi all. I'm trying to prove energy conservation in a (maybe) uncommon way. I know there are different ways to do this, but it is asked me to prove it this way and I'm stucked at the end of the proof. I'm considering ##N## bodies moving in a gravitational potential, such that the energy is ##E =...
  43. C

    I Proof check for Plancherel's theorem (Fourier Transform version)

    I'm trying to prove Plancherel's theorem for functions $$f\in L^1\cap L^2(\mathbb{R})$$. I've included below my attempt and I would really appreciate it if someone could check this for me please, and give me any feedback they might have. **Note:** I am working with a slightly different...
  44. B

    Proof for Cauchy sequences

    I've started by writing down the definitions, so we have $$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$ $$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w...
  45. N

    MHB What is the Epsilon-Delta Method for Proving Limits?

    Use the epsilon-delta method to show that the limit is 3/2 for the given function. lim (1 + 2x)/(3 - x) = 3/2 x-->1 I want to find a delta so that | x - 1| < delta implies |f(x) - L| < epsilon. | (1 + 2x)/(3 - x) - (3/2) | < epsilon -epsilon < (1 + 2x)/(3 - x) - 3/2 < epsilon I now add...
  46. Frabjous

    I Probability fun time: Proof that 1/3=1/2=1/4

    Forgive me, I am not a probability guy, so I am unsure how well known this is. I was trying to figure something out and found this. I found it cool. Here's the explanation. The first solution is a fraction (damn scanner!) Oops! From Kendall Geometrical Probability (1963)
  47. M

    MHB Logic Proof With Rules of Replacement

    Not sure if this is an allowed post, as it is not technically math but I'm trying to work through the below proof. If workers have a fundamental right to a job, then unemployment will be virtually nonexistent but job redundancy will become a problem. If workers have no fundamental right to a...
  48. larginal

    Proof of Electromagnetic Identity: Puzzling Last Expression

    I tried to understand proof of this identity from electromagnetics. but I was puzzled at the last expression. why is that line integral of dV = 0 ? In fact, I'm wondering if this expression makes sense.
  49. mcastillo356

    B Proof of Chain Rule: Understanding the Limits

    First I quote the text, and then the attempts to solve the doubts: "Proof of the Chain Rule Be ##f## a differentiable function at the point ##u=g(x)##, with ##g## a differentiable function at ##x##. Be the function ##E(k)## described this way: $$E(0)=0$$...
  50. C

    Indirect Proof Proof verification: sequence a_n=(−1)^n does not converge

    Theorem: Show that the sequence ## a_n = (-1)^n ## for all ## n \in \mathbb{N}, ## does not converge. My Proof: Suppose that there exists a limit ##L## such that ## a_n \rightarrow L ##. Specifically, for ## \epsilon = 1 ## there exists ## n_0 ## s.t. for all ## n > n_0## then ##|(-1)^n-L|<1##...
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