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

    Is the Proof for Normalization in Quantum Mechanics Valid?

    Homework Statement In Griffiths Introduction to Quantum Mechanics textbook, he shows that for any wave function that is time-dependent (which implies that the state of any particle evolves with time), the wave function will stay normalized for all future time. There is a step in the proof that...
  2. Zeeree

    Epsilon-delta proof for limits (multivariable)

    Homework Statement : the question wants me to prove that the limit of f(x,y) as x approaches 1.3 and y approaches -1 is (3.3, 4.4, 0.3). f(x,y) is defined as (2y2+x, -2x+7, x+y). [/B] The attempt at a solution: This is the solution my lecturer has given. it's not very neat, sorry...
  3. K

    I Understanding the Heine Borel Theorem: An In-Depth Analysis

    Hello, I have a question about Heine Borel Theorem. First, I am not sure why we have to show "gamma=Beta" gamma is the supremum of F(which is equivalent to H_squiggly_bar in the text ), and it has to be greater than beta. Otherwise, S contains H_squiggly_barSecond, for the case 1, why...
  4. A

    I Solving Mass-Speed Relation Proof Challenges

    I have a hard time understanding the variation of mass with velocity, more precisely the proof. In almost every material I've found, the author analyses 2 bodies colliding. The idea of looking at the collision is not hard to grasp and by considering one of the velocities equal zero, you get a...
  5. K

    MHB Proof: K is a Root Field for Every Irreducible Polynomial with a Root in K

    Suppose [K:F]=n, where K is a root field over F. Prove K is a root field over F of every irreducible polynomial of degree n in F[x] having a root in K. I don't believe my solution to this problem because I 'prove' the stronger statement: "K is a root field over F for every irreducible...
  6. JasMath33

    A Is This Proof Correct? Ask & Discuss Here

    I was reading this book yesterday and looking at this proof/justification. I was thinking it is possibly incorrect, but wanted to get some other opinions. Here is the example they gave in the book with the work attached.
  7. Alfreds9

    B Visual proof of being at altitude?

    Hi, this may seem like an odd questions to most of you but I'd still like to ask what could be some visual proofs of being at high altitude, say 10,000 feet above sea level. While any said proof is not extremely rigorous or untamperable and probably little more than a showy capture to add to...
  8. EternusVia

    How Does Sum of Consecutive Numbers Relate to Cubes of Integers?

    Homework Statement Consider the table: 1 = 0 + 1 2 + 3 + 4 = 1 + 8 5 + 6 + 7 + 8 + 9 = 8 + 27 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64 Guess the general law suggested by these examples, express it in suitable mathematical notation, and prove it. Homework Equations [/B] It's clear that if...
  9. P

    Tough geometry problem about triangles, proof

    Homework Statement let be ABC a generic triangle, build on each side of the triangle an equilater triangle, proof that the triangle having as vertices the centers of the equilaters triangles is equilater Homework Equations sum of internal angles in a triangle is 180, rules about congruency in...
  10. Math Amateur

    The Weak Nullstellensatz .... aspects of proof by Cox et al

    Homework Statement I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ... I am currently...
  11. Math Amateur

    MHB Understand Theorem 1: Weak Nullstellensatz Proof by Cox et al - Exercise 3(a)

    I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ... I am currently focused on Chapter 4...
  12. SamRoss

    I Proof of double angle formulas using Euler's equation

    Hi all, I'm slowly working through "Mathematical Methods in the Physical Sciences" by Mary Boas, which I highly recommend, and I'm stumped on one of the questions. The problem is to prove the double angle formulas sin (2Θ)=2sinΘcosΘ and cos(2Θ)=cos2Θ-sin2Θ by using Euler's formula (raised to...
  13. JasMath33

    Help Solve Calculus Limit Proof Homework Statement

    Homework Statement I am posting this for another student who I noticed did not have the proof in the problem. Here is what she said. Let's try and help her out. I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says x=0. If x-0, it...
  14. kyphysics

    Why Do People Say Accounting is Recession Proof? And is it?

    Very curious. Is there a supply and demand imbalance? When there is a recession and businesses are stagnant and new ones aren't starting up, how do accountants still get good work?
  15. D

    I Proving Isometries: A Step-By-Step Guide

    Hello, i'm trying to prove this statements, but I'm stuck. Be ##V=R^n## furnished with the standard inner product and the standard basis S. And let W ##\subseteq## V be a subspace of V and let ##W^\bot## be the orthogonal complement. a) Show that there is exactly one linear map ##\Phi:V...
  16. J

    Help with Epsilon Delta Proof of Multivariable Limit

    Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...
  17. J

    I Help With Epsilon Delta Proof of multivariable limit

    I realized I placed this in the wrong forum... I will put it in coursework help
  18. A

    Is there any absolute proof that photons exist?

    I mean, look this stupidity: [Mentor's note - link to crackpot site deleted] This guy denies that light photons exist, and that we are 'magically creating it' like cyclops X-Men This is worst than flat-earthers, I wonder If there is some evidence or is it unfalsifiable, like solipsism? Because I...
  19. R

    Deriving the Fourier Transform of the Signum Function and Proving its Properties

    Homework Statement The signum function is defined by$$sgn(t)=\left\{\begin{matrix}-1, \ t<0\\0, \ t=0 \\ 1, \ t>0 \end{matrix}\right.$$It has derivative$$\frac{d}{dt} sign(t) = 2 \delta(t)$$Use this result to show that ##j2\pi \nu S(\nu)=2,## and give an argument why ##S(0)=0.## Where...
  20. F

    B Vector Calculus Identity proof?

    The following identity is found in a book on Turbulence: Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki. Thanks
  21. Q

    I What is the proof for 1+4+9+16+....=0?

    Hello, I started to learn divergent series/sums, to practice I calculated some basic ones, you know: 1+2+3+4+5+6...= -1/12, but I really had problems when i tried to demonstrate that 1+4+9+16+...= 0(the sum of squares of natural numbers), I've tried to add, subtract etc, but I couldn't prove it...
  22. F

    A Proof for Close Packing of Congruent Identical Spheres

    I developed two algorithms for calculating the density of close packed congruent identical spheres in two different arrangements: A tetrahedron with four equilateral triangular faces, and A square pyramid with a square base and four equilateral triangular faces, as shown below. Figure...
  23. S

    MLE of Bivariate Vector Random Variable: Proof & Explanation

    Homework Statement Consider the bivariate vector random variable ##(X,Y)^T## which has the probability density function $$f_{X,Y}(x,y) = \theta xe^{-x(y+\theta)}, \quad x\geq 0, y\geq 0 \; \; \text{and} \; \; \theta > 0.$$ I have shown that the marginal distribution of ##X## is ##f_X(x|\theta)...
  24. Ma Xie Er

    A Why Is the Equality in This Spectral Analysis Proof Correct?

    I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification. It goes like this: Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...
  25. ubergewehr273

    Proving 1/3 < log_{34} 5 < 1/2

    Homework Statement Prove that ##{1/3} < log_{34} 5 < {1/ 2}## Homework Equations ##log_b a = {1/ log_a b}## ##logmn = logm + logn##The Attempt at a Solution ##log_{34} 5 = {1/ log_5 34}## ##= 1/(log_5 17 + log_5 2)## ##=1/(1 + log_5 3 + log_5 2 + something)##...
  26. Q

    I Why is E(t) multiplied by e^(-ix) in Plancherel's Theorem proof?

    the first step of the Plancherel's Theorem proof found in: http://mathworld.wolfram.com/PlancherelsTheorem.html, says: let be a function that is sufficiently smooth and that decays sufficiently quickly near infinity so that its integrals exist. Further, let and be FT pairs so that...
  27. V

    I Prove a(b-c)=ab-ac: Is It Enough?

    I am currently working my way though Calculus by Tom Apostol. One of the really early proofs ask the reader to prove: a(b-c)=ab-ac. Here is what I did, I let x=b-c which by the definition of subtraction equals x+c=b. Substituting that value into the right hand side I got...
  28. G

    Proof of image formation property of spherical mirrors

    Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same...
  29. T

    Proving theorem for polynomials

    Homework Statement Prove the following statement: Let f be a polynomial, which can be written in the form fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0 and also in the form fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0 Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n Homework Equations 3. The...
  30. C

    Capacitance vs. Resistance Proof

    Can somebody confirm if this is correct? I'm trying to use a wye-delta transformation on capacitors to solve for equivalent capacitance, but to be super-precise, I want to put capacitance in terms of resistance. I = C*(dV/dt) V = IR, so I = V/R V/R = C*(dV/dt) (V*dt) = R*C* dV Integrate both...
  31. M

    Ordered set proof review request

    Homework Statement , relevant equations, and the attempt at a solution are all in the attached file. I was reading through Invitation to Discrete Mathematics and attempted to solve an exercise that involved a proof. I've typeset everything in LaTeX and made a PDF out of it so that it does not...
  32. Memocyl

    Mathematical Proof of Kepler's First Law of Orbits

    Hello friends (I hope :biggrin:), For a maths project I am working on, I need to be able to prove the equation for an elliptical orbit, related to Kepler's first law: and p = a(1-e2) (or should be as p can be replaced by that value) Where: r = distance from sun to any point on the orbit p =...
  33. Danielm

    Proving the Bijectivity of a Function: σ : Z_11 → Z_11 | Homework Solution

    Homework Statement Let σ : Z_11 → Z_11 be given by σ([a]) = [5a + 3]). Prove that σ is bijective. Homework EquationsThe Attempt at a Solution I am just wondering if I can treat σ as a normal function and prove that is bijective by using the definitions of one to one and onto.
  34. Danielm

    Proving R = I_X: Equivalence Relation and Function Homework Solution

    Homework Statement Let X be a set and R ⊂ X × X. Assume R is an equivalence relation and a function. Prove that R = I_X, the identity function. Homework EquationsThe Attempt at a Solution Proof We know that R has to be reflexive, so for all elements b in X, bRb but b can't be related to any...
  35. Biker

    Electric Field Proof: Can Point A Have a Higher Field Than Point B?

    I had a thought about electric fields created by charges Look at this picture: Point ##B## is at the half the distance between ##q## and ##2q##. What I am trying to prove/disprove That there might be actually a point (##A##) near of charge ##2q## that might have an electric field stronger than...
  36. entropy1

    I Proof of unitarity of time evolution in Susskind's book

    In "The Theoretical Minimum" of Susskind (p.98) it says that if we take any two basisvectors |i \rangle and |j \rangle of any orthonormal basis, and we take any linear time-development operator U, that the inner product between U(t)|i \rangle and U(t)|j \rangle should be 1 if |i \rangle=|j...
  37. Danielm

    Proving Linear Independence and Spanning in Vector Spaces

    Homework Statement Prove the following: Let V be a vector space and assume there is an integer n such that if (v1, . . . , vk) is a linearly independent sequence from V then k ≤ n. Prove is (v1, . . . , vk) is a maximal linearly independent sequence from V then (v1, . . . , vk) spans V and is...
  38. Danielm

    Proof: Linear Dependence of Vectors in a Vector Space

    Homework Statement Prove the following theorem: Let (v1, . . . , vk) be a sequence of vectors from a vector space V . Prove that the sequence if linearly dependent if and only if for some j, 1 ≤ j ≤ k, vj is a linear combination of (v1, . . . , vk) − (vj ). Homework EquationsThe Attempt at a...
  39. D

    Mathematical proof (Drawing a help line)

    Homework Statement I'm doing quite a strict proof in school. Where we should proof something and use mathematical language and symbols. Homework Equations The Attempt at a Solution To proof what I have to proof I need to draw some help lines. As for instance the "red" one I did from A to B...
  40. B

    Postmodernists: proof of points of triviality and sophistry

    In Charles Murray's book Real Smart: Four Simple Truths For Bringing America's Schools Back To Reality, Murray writes about the postmodernists in literary criticism. His description really gets my interest. I think it would be interesting and perhaps amusing (I have a strange sense of humor)...
  41. D

    So the shortest title I can come up with is: Complex Numbers and Real Solutions

    Homework Statement z1, z2 are complex numbers. If z1z2 =/= -1 and |z1| = |z2| = 1 then number : z1 + z2 ________ 1 + z1z2 is real. Homework EquationsThe Attempt at a Solution z1 = (a+bi), z2 = (c+di)[/B] Should i use this extended form or is there a shorter...
  42. Danielm

    Counting Reflexive and Anti-Symmetric Relations on a Finite Set

    Homework Statement Let X = {1, 2, 3, 4, 5, 6}. Determine the number of relations on X which are reflexive and anti-symmetric Homework EquationsThe Attempt at a Solution This problem looks a little bit hard. Approach: consider R={(x,x),... } If there is just one pair in the relation in the...
  43. S

    B Proof that non-integer root of an integer is irrational

    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...
  44. I

    Limit points, closure of set (Is my proof correct?)

    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)...
  45. edguy99

    A Is there proof that black holes really exist?

    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...
  46. L

    Can Fermat's Little Theorem Help Disprove This Statement?

    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...
  47. L

    Proving Even Integer Coefficients in Quadratic Polynomials - Homework Question

    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...
  48. S

    The principle of least Action proof of minimum

    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...
  49. M

    MHB Can Logarithmic Functions Be Expressed as Infinite Series?

    Hi, I'm stuck on the following proof: \log[3] = \frac 1{729} \sum_{k=0}^\infty \frac 1{729^k} \left[\frac{729}{6k+1}+\frac{81}{6k+2}+\frac{81}{6k+3}+\frac 9{6k+4}+\frac 9{6k+5}+\frac 1{6k+6}\right] Manipulating and converting summands to integrals of the form $x^{-(6k+n)}$ over {x,0,3} seems...
  50. L

    Is 3^n Greater Than n^4 for All n>=8?

    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...
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