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. C

    B Is there any proof that gravity is an attractive force?

    Is gravity an attractive force of matter, or is it a universal force of expansion and pressure that exists in apparently empty space? Does matter attract objects near it, or does matter merely block gravity pressure on on any given side of a material body, giving the illusion that the matter...
  2. S

    Proof of (x^n - 1) / (x - 1) = x^n-1 + x^n-2 + .... + x + 1

    Homework Statement The problem comes from Lang's Basic Mathematics, chapter 1, paragraph 6 (multiplicative inverses) and simply asks to prove the relation: (xn - 1) / (x - 1) = xn - 1 + xn - 2 + ... + x + 1 Homework Equations a-1a = aa-1 = 1 Cross-multiplication rule Cancellation law for...
  3. L

    B Product of Negative Real Numbers: A Geometric Proof

    What is the Geometric Proof for the product of two negative real numbers being a positive real number?
  4. Mr Indeterminate

    I Where has this proof gone wrong? ∞= 1/0

    Now I expect that most of you on this forum would be familiar with the equality between point nine reoccurring and one: 0.999...=1 If your not familiar please review https://en.wikipedia.org/wiki/0.999... Now this equality can be used to imply something else, which is rather heterodox...
  5. M

    Is there any proof to show that gravity works at the Planck length?

    Is there any experimental proof which shows that the force of gravity works at Planck length? Has it been verified already? Is it verifiable?
  6. evinda

    MHB Questions about proof about primality testing

    Hello! (Wave) I have read the proof that primality testing is in NP and I have a question about it. The proof is the following:Note that the group $(\mathbb{Z}/ N \mathbb{Z})^{\ast}$ is of order $N-1$ iff $N$ is prime. And more over, it is a cyclic group of order $N-1$ iff $N$ is a prime...
  7. S

    B Vector Proof: Solving for IuI and IvI using Dot Product Properties

    I am having trouble with this proof. I just need a step in the right direction. Let u and v be vectors. (u+v)*(u-v)=0, then IuI=IvI I have to use properties of the dot product. I started off by combining both using this property u*(v+w)=u*v+u*w (u,v,w are vectors) I got lost in all of my...
  8. F

    Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture | Polynomial Homework

    Homework Statement Prove or refute the following conjecture: There are no positive integers x and y such that ##x^2 - 3xy + 2y^2 = 10## Homework Equations ##10 = 5*2## ##10 = 10*1## The Attempt at a Solution I graphed it using a graphing calculator, so I know this is true. Proof: This will...
  9. RJLiberator

    Reciprocal Theorem proof in 3 variables

    Homework Statement w is a function of three variables x, y, and z. Prove that \frac{\partial w}{\partial x}_{y,z} = \frac{1}{\frac{\partial x}{\partial w}}_{y,z} Homework EquationsThe Attempt at a Solution w=w(x,y,z) dx = \frac{\partial w}{\partial x}_{y,z}dx +\frac{\partial w}{\partial...
  10. Mr Davis 97

    I Addition of exponents proof in group theory

    Just out of curiosity, what would a proof of ##a^m a^n = a^{m+n}## amount to? Of course obviously if you have n of one thing and m of another you get m+n, but I am wondering if this is rigorous enough, or if you need induction.
  11. T

    I How can -a be equal to (-1)*a?

    I will say that this question is coming from a lack of explanation in a classroom, however this particular proof is not homework and is just explanation over a proof that was discussed briefly in class, so I didn't think it belong in the homework section. I'm also not certain it belongs in the...
  12. C

    Purely Inductive Circuit -- Mathematical proof for current lag

    how we can mathematically prove that in a purely inductive circuit current lags behind voltage by a phase angle of π/2?
  13. Mr Davis 97

    Circle Geometry Proof: Perpendicular Chord Bisected by Diameter

    Homework Statement Prove that any chord perpendicular to the diameter of a circle is bisected by the diameter. Homework EquationsThe Attempt at a Solution I was thinking that maybe I could form two triangles, show that these triangles are congruent, and then conclude that the two lengths of...
  14. Math Amateur

    MHB Proof of Cauchy's Inequality .... Sohrab Proposition 2.1.23 ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Proposition 2.1.23/Exercise 2.1.24 (Exercise 2.1.24 asks readers to prove Proposition 2.1.23) ... Proposition...
  15. Math Amateur

    Proof of Cauchy's Inequality .... Sohrab Proposition 2.1.23

    Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Proposition 2.1.23/Exercise 2.1.24 (Exercise 2.1.24 asks readers to prove Proposition 2.1.23)...
  16. R

    Courses Is proof based Linear Algebra be similar to Abstract Algebra

    I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract...
  17. D

    Prove trace of matrix: Tr(AB) = Tr(BA)

    Homework Statement [/B] The trace of a matrix is defined to be the sum of its diaganol matrix elements. 1. Show that Tr(ΩΛ) = Tr(ΩΛ) 2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic) my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U...
  18. vantroff

    B (Proof) Two right triangles are congruent.

    Hi, the question is from Serge Lang - Basic mathematics, Page 171 exercise 6. Thing to prove: Let ΔPQM and ΔP'Q'M' be right triangles whose right angles are at Q and Q', respectively. Assume that the corresponding legs have the same length: d(P,Q)=d(P',Q') d(Q,M)=d(Q',M') Then the right...
  19. Z

    Calculus Please recommend a textbook containing a rigorous proof of ....

    This is excerpted from George E. Hay's "Vector and Tensor Analysis". The author gives the statement that the limit is equal to 1 without any explanation, perhaps because he thinks it does not belong to the contents of vector analysis. I can see it intuitively, but I want a rigorous mathematical...
  20. binbagsss

    Wick theorem proof 4 point correlator 2 different fields

    1. Homework Statement Attached: Homework Equations I've just changed the notation a tad to make things quicker for me : ##\phi_1=\phi_1(x_1)## and ##\Phi_2=\phi_2(x_2)## : denotes normal product. i.e annhilator operators are on the RHS, so acting on a vacuum state will give zero. I can...
  21. F

    Why Does a Prime Number Divide Its Binomial Coefficients?

    Homework Statement The book wants me to use direct proof. if p is a prime and k is an integer for which 0 < k < p, then p divides ##\left( \frac p k \right)## Homework Equations ##\left( \frac p k \right) = \frac {p!} {k!(p-k)!}## The Attempt at a Solution the fraction line in ##\left( \frac...
  22. A

    Spivak chapter 3 problem 24 - proof of a composition

    Homework Statement Suppose g is a function with the property that g(x) =/= g(y) if x=/=y. Prove that there is a function f such that f( g(x) ) = x. (The composition) Homework Equations Definition of a function, collection of ordered pairs; g(x) =/= g(y) if x=/=y; x → g(x) → x (The composition...
  23. Mr Davis 97

    I Proof that the square root of 2 is irrational

    Quick question: In the proof that the square root of 2 is irrational, when we are arguing by contradiction, why are we allowed to assume that ##\displaystyle \frac{p}{q}## is in lowest terms? What if we assumed that they weren't in lowest terms, or what if we assumed that ##\operatorname{gcd}...
  24. Derek Hart

    B Proof that (x^n)/n has a limit of 0 at infinity

    I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF: Firstly, I assume that x is positive. I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...
  25. F

    I Representations of the Poincaré group: question in a proof

    Hello! :smile: On page 51 where he want to invert $$\Lambda^{\mu}_{\nu} = \tfrac{1}{2} \text{tr}( \bar{\sigma}^{\mu}A \sigma_{\nu} A^{\dagger})$$ the person says we may use $$\sigma_{\nu} A^{\dagger} \bar{\sigma}^{\nu} = 2 \text{tr}(A^{\dagger})I.$$ to do that ... how do you prove this formula...
  26. binbagsss

    Proof of Wick's Theorem for 3 fields

    Homework Statement Question attached: Homework Equations [/B] Using the result from two fields that ## T(\phi(x) \phi(y))= : \phi(x) \phi(y) : + G(x-y)## Where ##G(x-y) = [\phi(x)^+,\phi(y)^-] ## ## : ## denotes normal ordered and ##\phi(x)^+ ## is the annihilation operator part , and...
  27. F

    Proof of irrationality of sqrt2 in Penroses Road to Reality

    Hello Everybody, I hope that I've picked the right sub-forum for my question. The problem is as follows: I understand perfectly well the proof by contradiction of the irrationality of the square root of 2, where we prove that if we assume that it is rational, then the integers of which the...
  28. G

    A Usage of kinetic energy in proof for E=mc^2

    http://bildr.no/view/aWc4dW95 Above is a question that I posted on a school help site. This is the answer I got: https://www.scribd.com/document/352723366/problem-about-usage-of-E-mc-2-for-v-much-smaller-then-c-ans Can you use this answer to show that the type of calculation that I tried in...
  29. S

    Hyperbolic Geometry (Rectangles)

    Homework Statement Recall that in hyperbolic geometry the interior angle sum for any triangle is less than 180◦. Using this fact prove that it is impossible to have a rectangle in hyperbolic geometry. Homework EquationsThe Attempt at a Solution - I wanted to use the idea that rectangles are...
  30. binbagsss

    Elliptic functions proof f(z)-c has N zeros, N the order

    Homework Statement proof of theorem Homework Equations The Attempt at a Solution Hi, I have a couple of questions on the attached proof and theorem 1) On the last line, how is it we go from the order of the zeros = the number of zeros, or is it's meaning the number of zeros counted with...
  31. Adgorn

    Proof regarding the image and kernel of a normal operator

    Homework Statement Show that if T is normal, then T and T* have the same kernel and the same image. Homework Equations N/A The Attempt at a Solution At first I tried proving that Ker T ⊆ Ker T* and Ker T* ⊆ Ker, thus proving Ker T = Ker T* and doing the same thing with I am T, but could not...
  32. Adgorn

    Proving properties of a 2x2 complex positive matrix

    Homework Statement Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\ c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc## Homework Equations N/A The Attempt at a Solution I got stuck at the first part. if ##A## is positive...
  33. S

    Proving Independence of Fano's Geometry Axiom 4

    Homework Statement In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one...
  34. S

    I Understanding QM Proof: Wavefunction in Orthonormal Eigenfunctions

    Hello! I have a proof in my QM book that: ##\left<r|e^{-iHt}|r'\right> = \sum_j e^{-iHt} u_j(r)u_j^*(r')##, where, for a wavefunction ##\psi(r,t)##, ##u_j## 's are the orthonormal eigenfunctions of the Hamiltonian and ##|r>## is the coordinate space representation of ##\psi##. I am not sure I...
  35. SamRoss

    B Proof of quotient rule using Leibniz differentials

    I know how to prove the quotient rule by using the definition of a derivative using limits (Newton's style). I just saw a proof of the product rule using Leibniz's concept of differentials on Wikipedia. https://en.wikipedia.org/wiki/Product_rule#Discovery Does anyone know of a Leibniz-style...
  36. SamRoss

    B Don't follow one small step in proof

    https://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/inverse_trig The above link is just a proof for the derivative of arcsinx. Going from line 3 to 4, how does d/dx siny become dy/dx cosy ?
  37. M

    Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction

    Homework Statement Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction. Homework Equations None The Attempt at a Solution The base case which is when ##n=2##: ##5^2+9<6^2## ##34<36## Thus, the base case is true. Now for the induction step. Induction hypothesis: Assume...
  38. B

    B Understanding Invertible Matrices and Homogenous Systems

    For a ##n\times n## matrix A, the following are equivalent. 1) A is invertible 2) The homogenous system ##A\bf X = 0## has only the trivial solution ##\mathbf X = 0## 3) The system of equations ##A\bf X = \bf Y## has a solution for each ##n\times 1 ## matrix ##\bf Y##. I have problem in third...
  39. B

    B Proof of elementary row matrix operation.

    Prove that interchange of two rows of a matrix can be accomplished by a finite sequence of elemenatary row operations of the other two types. My proof :- If ##A_k## is to be interchanged by ##A_l## then, ##\displaystyle \begin{align} A_k &\to A_l + A_k \\ A_l &\to - A_l \\ A_l &\to A_k + A_l...
  40. F

    Proving the Evenness of Elements Not Equal to Their Own Inverse in Finite Groups

    Homework Statement Prove in any finite group G, the number of elements not equal to their own inverse is an even number. Homework Equations if ab = ba = e, then a = b-1 and b = a-1 The Attempt at a Solution Let S, A, B, be subsets of G where S = A + B. Let a ∈ A s.t. there exists a unique b...
  41. stevendaryl

    I Prove Inequality: A,A', B, B' in [0,1]

    I'm pretty sure that the following is true, but I don't see an immediate compelling proof, so I'm going to throw it out as a challenge: Let A,A', B, B' be four real numbers, each in the range [0,1]. Show that: AB + AB' + A'B \leq A' B' + A + B (or show a counter-example, if it's not true)...
  42. PHstud

    Proof of fundamental thermodynamics equation for open systems

    Hi ! I'm having a bit of trouble understanding something. Let 'u' be internal energy, 'h' enthalpy, 'e' work and 'q' heat. ('r' are dissipations and 'S' entropy) From a book , i read that de+dr=PdV= -du + TdS This seems to stand for closed cycle. Yet, my teacher uses the formula de+dr=vdP=...
  43. K

    B What is the role of tautology in an axiomatic system and proof?

    Hi, I have a question about axiomatic system and proof. Let's say we have a finite sequence of propositions ai, which is an axiomatic system. To prove a proposition P that is a finite sequence of propositions qi with axiomatic system {ai}, we can take 3 methodologies. (A) qi itself is...
  44. S

    I Proof of Harmonic Function Infinitely Differentiable

    Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof...
  45. B

    Solving Symmetric Group Induction Proof: Hints for Double Induction

    Homework Statement Consider the symmetric group ##S_n##. I am trying to establish that ##(i,i+1)=(1,2,...,n)(i-1,i)(1,2,...,n)^{-1}## Homework EquationsThe Attempt at a Solution I am trying to decide whether I need double induction or not. I have done several calculations to see whether I can...
  46. Adgorn

    Proof regarding transpose mapping

    Homework Statement Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1. Homework Equations N/A The Attempt at a Solution Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So...
  47. SherLOCKed

    A Help with proof of eq. 2.64 of Intro. to Quantum Mechanics

    I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47. While the book has given a proof for eq. 2.64 but its not very ellaborate Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody...
  48. P

    Need help understanding the proof of Thevenin's theorem

    Hello. Uh, I'm trying to undestand how to prove Thevenin's theorem. The Sadiku book puts an independent current source where the load used to be in order to reach the equation: V = Vth + I*Rth. I do understand how he reaches that conclusion after putting the source, what I don't understand is...
  49. G

    I Is the Proof of Geometric Progression in Probability Common Sense?

    My Statistics textbook does not prove this. The author think it is commons sense. I am not sure about this proof. Thank you.
  50. Adgorn

    Proof regarding linear functionals

    Homework Statement Let V be a vector space over R. let Φ1, Φ2 ∈ V* (the duel space) and suppose σ:V→R, defined by σ(v)=Φ1(v)Φ2(v), also belongs to V*. Show that either Φ1 = 0 or Φ2 = 0. Homework Equations N/A The Attempt at a Solution Since σ is also an element of the duel space, it is...
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