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. M

    Proof of Subgroup Property for Cyclic Group G: Homework Help

    Homework Statement Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G. I am confused as to why the proof makes the assumption that <a> is a subset of the set G. Homework EquationsThe Attempt at a Solution The proof I think is like the...
  2. M

    I Question regarding a sequence proof from a book

    I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
  3. T

    I Proof Explanation: Showing an extension to a continuous function

    I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
  4. B

    Mathematical Analysis Proof: |x-y|<= |x|+|y|

    Homework Statement 1. Show that for all real numbers x and y: a) |x-y| ≤ |x| + |y| Homework Equations Possibly -|x| ≤ x ≤ |x|, and -|y| ≤ y ≤ |y|? The Attempt at a Solution I tried using a direct proof here, but I keep getting stuck, especially since this is my first time ever coming...
  5. E

    I Spivak's proof of Cauchy Schwarz

    I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...
  6. B

    Courses Applied vs Proof Based Linear Algebra

    Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of...
  7. T

    I Proof that cube roots of 2 and 3 are irrational

    Proof by contradiction that cube root of 2 is irrational: Assume cube root of 2 is equal to a/b where a, b are integers of an improper fraction in its lowest terns. So the can be even/odd, odd/even or odd/odd. The only one that can make mathematical sense is even/odd. That is...
  8. S

    I What is the proof that the divergence is normal to the surface?

    If I am given a function f( x , y , z , ...) = C then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?
  9. S

    Studying Physics students and proof based calculus

    Hey, I have been told to study calculus following Spivak's book. I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it. The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the...
  10. Math_Maniac

    Moment of Inertia of Solid Sphere - Proof

    So I have been having a bit of trouble trying to derive the moment of inertia of a solid sphere through its center of mass. Here is my working as shown in the attached file. The problem is, I end up getting a solution of I = (3/5)MR^2, whereas, in any textbook, it says that the inertia should...
  11. S

    I Difference between Constructive proof and Existential Generalization?

    What is the difference between Constructive Proof of existence and Existential generalization? Logically they seem to be the same because, for a given predicate and specific member of the predicate's domain, you are concluding the general statement about the predicate.
  12. I

    [Linear Algebra] Linear transformation proof

    Homework Statement Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace. (a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B## (b) Let ##C \subseteq V## be a...
  13. Mathmellow

    MHB Understanding induction proof of pigeonhole principle

    I am struggling to understand the induction proof of the pigeonhole principle in my textbook. The theorem and the proof, from Biggs Discrete Mathematics, is pasted below, and I will explain further (see bold text) what I am having trouble with. Theorem. Let m be a natural number. Then the...
  14. T

    I Zorn's Lemma: Need help finding errors in proof

    Proposition(Zorn's Lemma): Let ##X\neq\emptyset## be of partial order with the property that ##\forall Y\subseteq X## such that ##Y## is of total-order then ##Y## has an upperbound, then ##X## contains a maximal element. Proof: Case 1: ##B\neq\emptyset## such that ##B##=##\{####b\in X##: ##b##...
  15. R

    Intro Math What to read after "Book of Proof?"

    Hi since U.S. education is shite, I've decided that I'm going to learn math from the ground up by myself. My goal is to reach graduate level mathematics in 2-3 years. I'm currently reading Book of Proof, what should I read after this? My end goal is to be proficient in applied math/ physics.
  16. Philethan

    A Bloch theorem proof with V(x)=V(x+ma)

    In Grosso's Solid State Physics, chapter 1, page 2, The author said that: Therefore, I plug (4) into (1), and I expect that I can get the following relationship, which proves that ##H\left|W_{k}(x)\right\rangle## belongs to the subspace ##\mathbf{S}_{k}## of plane waves of wavenumbers...
  17. J

    A Does the Frauchiger-Renner Theorem prove only MWI is correct

    Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page. I am asking in reference to this paper: https://arxiv.org/pdf/1604.07422.pdf Which claims to show that single...
  18. binbagsss

    Quick question on Laurent series proof uniqueness

    Homework Statement I am looking at the wikipedia proof of uniqueness of laurent series: https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations look above or belowThe Attempt at a Solution I just don't know what the indentity used before the bottom line is, I've never seen it before...
  19. binbagsss

    Elliptic Functions Proof of Sum of Residues=0

    Homework Statement Hi I am looking at the attached proof for this property. I agree with the first line due to periodicity, but unsure about the next- see below 3)attempt Homework Equations To me, I deemed the integration substituion rule as relevant to this question, but perhaps...
  20. T

    I How can I go about making a "Space Proof" coating?

    So I am planning on launching a Satellite to promote the Dogecoin cryptocurrency. One of the main points is printing/painting (Or whatever) the logo on the side of a metal panel. How can I make it so it doesn't melt off or turn white from radiation so quickly?
  21. B

    Proof of oscillation about the equilibrium

    Homework Statement The problem is question 2(a) in the attached pdf. I seem to find myself at a dead end and am not sure where to go from here - I will attach my working in a separate file, but basically I need to show that the oscillator passes/crosses over the x = 0 boundary at a positive...
  22. Mr Davis 97

    I Understanding Proof of Uniqueness

    I'm trying to really get a grasp on proofs of uniqueness. Here is a model problem: Prove that ##x=-b/a## is the unique solution to ##ax+b=0##. First method: First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##. Now, we show uniqueness: If ##ax+b=0##, then...
  23. Mr Davis 97

    I Understanding a Graph Theory Proof

    Prove that if a simple graph G has 6 vertices then G or its complement has a subgraph isomorphic to ##K_3##. The proof begins by noting that is must be the case that G or its complement as a vertex with degree at least 3. Why is this the case?
  24. isukatphysics69

    Not understanding calc proof of series

    Homework Statement Homework EquationsThe Attempt at a Solution I don't understand why for the first part where the series goes up until arn-1, it cannot just go up until arn.. why will that first series always go up until arn-1 until it is multiplied by r?
  25. Cheesycheese213

    B Fermat's little theorem proof?

    So I was taught that If gcd (a, p) = 1, then ap-1 ≡ 1 (mod p) And then the proof was Lemma: Let p be prime, Let i, j ,k = Integers If gcd (k, p) = 1 and ik ≡ jk (mod p) then i ≡ j (mod p) Main Proof: Consider 1a, 2a, 3a, ..., (p - 1)a Taking mod p is some arrangement of 1, 2, 3, ..., p - 1 Then...
  26. A

    I Proof that 1 is an odd number using Peano Axioms of naturals

    So I was just writing a proof that every natural number is either even or odd. I went in two directions and both require that 1 is odd, in fact I think that 1 must always be odd for every such proof as the nature of naturals is inductive from 1. I am using the version where 1 is the smallest...
  27. Mr Davis 97

    I Proof of Alternating Series Test

    I'm looking at the proof of the alternating series test, and the basic idea is that the odd and even partial sums converge to the same number, and that this implies that the series converges as a whole. What I don't understand is why the even and odd partial sums converging to the same limit...
  28. Mr Davis 97

    Epsilon-delta continuity proof

    Homework Statement Prove that ##f(x) = \frac{1}{x}## is continuous using the epsilon-delta definition of continuity. Homework EquationsThe Attempt at a Solution We will assume that the domain of ##f## is ##\mathbb{R} / \{ 0\}##. Let ##x_0## be in the domain. First, we look at ##\displaystyle...
  29. alexmahone

    MHB Could there be an error in the proof of the Poincare conjecture?

    When Grisha Perelman submitted his proof of the Poincare conjecture, he may have been reasonably sure that it contained no mistakes. But he could not have been 100% sure as he is, after all, human. Each time it was checked, say by the referee of an academic journal, the probability that it...
  30. cobalt124

    I Proof that BB(k) grows faster than any computable function

    Hi, layman post, not sure what thread level I need. From post https://www.physicsforums.com/threads/the-busy-beaver-function.942741/, I've been working my way through https://www.scottaaronson.com/busybeaver.pdf and come to section "1.3 The Busy Beaver Function", which states: "The Busy Beaver...
  31. M

    Logic Behind a Proof: Injective Function G

    Homework Statement suppose I have a function defined as: G: ℚ--->ℚ f(x)= { 2/ 3x if x does not equal to 0, 0 if x=0} Homework Equations Injective:if for all x,y in ℚ, f(x)=f(y) then x=y. or if x does not equal to y then f(x) does not equal to f(y)The Attempt at a Solution I am confused as to...
  32. rocdoc

    I Proof of Kaku (8.18): Completing the Square and Using Spiegel's Result

    In the following there is a proof, for positive values of ##a## only, of (8.18) of Kaku, reference 1, I quote' $$\int_{-\infty}^\infty~\mathrm{d}p~e^{iap^2+ibp}=\sqrt \frac{i\pi}{a}e^{-ib^2/4a}~~~~~~~~~~~~~(8.18)$$ '. Kaku says this result can be proved by completing the square. $$iap^2+ibp =...
  33. K

    I Proof of a law versus proof of a theorem

    if I get proof of fundamental laws like Newton's laws of motion or fundamental laws of thermodynamics then will they be laws anymore or will they become theorem. Please tell
  34. M

    Is this Proof of Equality Correct?

    Homework Statement Prove the following: If x=y and y=z then x=z. Now, this seems very obvious, and it is without a doubt correct. However, I am curious as to if the following proof is correct. Homework EquationsThe Attempt at a Solution Assume x does not equal to z, so that means two cases...
  35. Mr Davis 97

    I Proof that a sequence has two subsequential limits

    Suppose I have the sequence ##a_n = 2^{(-1)^n}##. So ##\displaystyle (a_n) = (\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,...)##. Clearly, this sequence has two subsequential limits, ##\displaystyle \{\frac{1}{2},2 \}##. This clear from observation, but I'm not sure how I can be sure...
  36. Mr Davis 97

    I Proof that a quantity is greater than 1/2

    I'm looking at the quantity ##\displaystyle 1 - \frac{N}{n}##, and trying to prove that it is greater than ##1/2##, given that ##n> N##. I thought that since ##\lim_{n \to \infty} 1 - \frac{N}{n} = 1##, we could use the definition of convergence to get this inequality, for suitable ##\epsilon##...
  37. R

    When is the minimum polynomial of a scalar matrix kI equal to t-k?

    Homework Statement Show that A is a scalar matrix kI if and only if the minimum polynomial of A is m(t) = t-k Homework EquationsThe Attempt at a Solution f(A) is monic f(A) = 0 since A = kI Next we must show that deg(g) < deg(f) I guess I'm not sure where g comes from. Is it merely an...
  38. jamalkoiyess

    I Proof that p is interior if p is not limit of complement

    Hello PF, I am searching for a proof that I couldn't find on the internet. Theorem: E in X a metric space. p in E. p is an interior point of E if and only if p is not a limit point of (E complement)' Sorry for notations but I have no idea how to insert Latex here.
  39. Mr Davis 97

    I Clarification of a line in a proof

    This comes from a line of a proof in my book, and I need help resolving why the equality is true. Suppose that ##M>N##. Why is it true that ##\displaystyle \sup \{\frac{1}{n} (s_{N+1} + \cdots + s_n) ~|~ n>M \} = \frac{n-N}{n}\sup \{s_n ~|~ n > N \}##?
  40. Z

    Help with Newton root approximation proof

    Homework Statement Suppose we have: ## f(x) = x^2 - b ## ## b > 0 ## ## x_0 = b ## And a sequence is defined by: ## x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i) } ## prove ## \forall i \in N ( x_i > 0 ) ## Homework Equations The Attempt at a Solution a)Here I tried solving for ## x_1 ## as...
  41. Physiona

    What is the algebraic proof for the remainder of 11 when dividing by 12?

    I'm currently doing a grade 9 paper, and one of the following questions is tripping me up a little bit: Prove algebraically that the sum of the squares of any three consecutive odd numbers always leaves a remainder of 11, when divided by 12. My attempt of the question: I have labelled 3...
  42. J

    I How Can Jensen's Inequality Be Used to Prove a Vector Magnitude Relationship?

    I have a vector B of length N, I would like to prove that: ∑n=0 to N-1 (|Bn|x) ≥ Nαx where: x > 1; α = (1/N) * ∑n=0 to N-1 (|Bn|) (i.e., The mean of the absolute elements of B). and ∑n=0 to N-1 (||Bn|-α|) ≠ 0; (i.e., The absolute elements of B are not all identical). I believe the above to...
  43. J

    MHB Polynomial Proof: Verification & Correction

    I would like to have verification if the following attached proof is correct. If it is not correct, what can be done to make it correct? Thanks.
  44. C

    MHB The proof of the infinite geometric sum

    Dear Everybody, I need some help with find M in the definition of the convergence for infinite series. The question ask, Prove that for $-1<r<1$, we have $\sum_{n=0}^{\infty} r^n=\frac{1}{1-r}$. Work: Let $\sum_{n=0}^{k} r^n=S_k$. Let $\varepsilon>0$, we must an $M\in\Bbb{N}$ such that $k\ge...
  45. barcodeIIIII

    Proof: Time independence of the entropy under unitary time evolution

    Homework Statement The unitary time evolution of the density operator is given by $$\rho(t)=\textrm{exp}(-\frac{i}{\hbar}Ht)\,\rho_0 \,\textrm{exp}(\frac{i}{\hbar}Ht)$$ General definition of entropy is $$S=-k_B\,Tr\,\{\rho(t) ln \rho(t)\}$$ Proof: $$\frac{dS}{dt}=0$$ Homework Equations I am not...
  46. O

    Is it possible to prove (P→Q)↔[(P ∨ Q)↔Q] without using truth tables?

    Homework Statement Need to demonstrate this proposition: (P→Q)↔[(P ∨ Q)↔Q] . My textbook use truth tables, but I'd like to do without it. It asks me if it's always truthThe Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right side...
  47. F

    Proving the Relationship between Lim Sup and Lim Inf

    Homework Statement Prove the ##\limsup \vert s_n \vert = 0## iff ##\lim s_n = 0##. Homework Equations ##\limsup s_n = \lim_{N\rightarrow \infty} \sup \lbrace s_n : n > N \rbrace = \sup \text{S}## ##\liminf s_n = \lim_{N\rightarrow \infty} \inf \lbrace s_n : n > N \rbrace = \inf \text{S}##...
  48. Mr Davis 97

    Real Analysis: Prove Upper Bound of Sum of Bounded Sequences

    Homework Statement Suppose that ##( s_n )## and ## (t_n)## are bounded sequences. Given that ##A_k## is an upper bound for ##\{s_n : n \ge k \}## and ##B_k## is an upper bound for ##\{t_n : n \ge k \}## and that ##A_k + B_k## is an upper bound for ##\{s_n + t_n : n \ge k \}##, show that ##\sup...
  49. Mr Davis 97

    Proof that a recursive sequence converges

    Homework Statement Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges. Homework EquationsThe Attempt at a Solution First, we must prove that the sequence is bounded below. We will prove that it is bounded below by 0. ##t_1 = 1 \ge 0##, so the base case...
  50. mertcan

    Bland rule proof linear programming

    <Moderator's note: Continued from a technical forum and thus no template. Re-opening has been approved by moderator.> Hi, my question is related to simplex algorithm anticycling rule called Bland's rule. While I was working with the proof in the link...
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