What is Proof: Definition and 1000 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. C

    Proof that T is bounded below with ##inf T = 2M##

    My first solution is Let ##S = \{x_1, x_2, x_3, ..., x_n\}## ##T = \{2x_1, 2x_2, 2x_3, ... 2x_n\}## ##T = 2S## Therefore, ##inf T = inf 2S = 2inf S = 2M## May someone please know whether this counts as a proof? My second solution is, ##x ≥ M## ##2x ≥ 2M## ##y ≥ 2M## (Letting y = 2M) Let...
  2. C

    Proof using Supremum

    For this problem, My solution: Using definition of Supremum, (a) ##M ≥ s## for all s (b) ## K ≥ s## for all s implying ##K ≥ M## ##M ≥ s## ##M + \epsilon ≥ s + \epsilon## ##K ≥ s + \epsilon## (Defintion of upper bound) ##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum ##M ≥ s +...
  3. C

    B Question about the fundamental theorem of calculus

    Hello everyone, I've been brushing up on some calculus and had some new questions come to mind. I notice that most proofs of the fundamental theorem of calculus (the one stating the derivative of the accumulation function of f is equal to f itself) only use a limit where the derivative is...
  4. M

    I Question about branch of logarithms

    I've read a proof from Complex Made Simple (David C. Ullrich) Proposition 4.3. Suppose that ##V## is an open subset of the plane. There exists a branch of the logarithm in ##V## if and only if there exists ##f \in H(V)## with ##f'(z) = \frac{1}{z}## for all ##z \in V##. Proof: One direction is...
  5. A

    Number of Multiplications in the FFT Algorithm

    Hello everyone, maybe some of you know the formula for the number of multiplications in the FFT algorithm. This is again given as ##N/2 \cdot log(N)##. Why is that so? Can you really "prove" this? I can only deduce this from what I know, because we have ##log(N)## levels and ##N/2##...
  6. C

    B Question about change of variables

    Hello everyone, I found a good proof for the area of a circle being ##{\pi}r^2## but I was recently working on my own proof and I used a change of variables and was wondering if I did it correctly and why a change of variables seems to work. I start with the equation of a circle ##r^2 = x^2 +...
  7. fluidistic

    Zero K proof that a chess position contains a checkmate

    Hi people, It's been years I wanted to post this question here. I would like to build a zero knowledge proof that a given chess position contains at least one checkmate. I know that anything provable admits a zero k proof. I know about...
  8. MAXIM LI

    I Determination of error in interpolating polynomial

    Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book where is proved but I failed. This is the theorem: Let $$ x_0 < x_1 < \cdots < x_n \in [a, b]...
  9. jedishrfu

    B What theorems are available when using modulo arithmetic?

    I'm looking for theorems related to using modulo arithmetic. As an example, if I apply a sequence of arithmetic operations to a given number to get an answer and then apply a modulo operation on the result to get a remainder in a given base. Wiil that be the same if I apply the modulo operation...
  10. G

    The Divergence of the Klein-Gordon Energy-Momentum Tensor

    I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough: $$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$ apply product rule to all terms $$=\partial^\nu \phi \cdot...
  11. Delta2

    Integration by factors

    I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself. Any helpfull tips?
  12. BugKingpin

    I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)

    Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H? My thinking is to show that ##N## and H satisfy...
  13. graphking

    I A strict proof of "why the Earth is a ball"

    "bubbles are ball" is called isoperimetric problem in serious mathematic. In this topic, many essay were written. Here's my serious essay about "why earth ball", which has been rejected by arxiv and my mentors...... I would want to know if physicists are interest? I really think that is...
  14. L

    Show that the limit (1+z/n)^n=e^z holds

    Hi, I have problems proving task d I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
  15. I

    Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates

    with this background, we proceed to the proof. Let us define a set $$ G = \{ z \in \mathbb{N} | \; x, y \in \mathbb{N}\; (x \cdot y) \cdot z = x \cdot (y \cdot z) \} $$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above...
  16. I

    Prove ##a\cdot b = b \cdot a ##using Peano postulates

    with this background, we proceed to the proof. Let us define a set $$ G = \{ z \in \mathbb{N} | \mbox{ if } y \in \mathbb{N}, y\cdot z = z \cdot y \} $$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above. Obviously, ## G...
  17. I

    Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates

    I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1...
  18. I

    Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##

    I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book) There exists a set...
  19. I

    Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##

    I have to prove that ##1 + a = s(a) = a + 1## using Peano postulates if ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book)...
  20. M

    Proof in number theory: the sum of all divisors of n

    let n be a positive integer show that if n is square then σ(n)( the sum of all divisors of n )is odd.
  21. L

    Proving limits for roots and exponents

    Hi I have to prove the following three tasks I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$ $$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...
  22. L

    Induction with binomial coefficient

    Hi, I'm having problems with the proof for the induction of the following problem: ##\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1} \binom{n}{k}=\frac{1}{n+1}## with ##n \in \mathbb{N}## I proceeded as follows: $$\sum\limits_{k=0}^{n+1} \frac{(-1)^k}{k+1} \binom{n+1}{k}=\frac{1}{n+2}$$...
  23. L

    Proving the Infimum and Supremum: A Short Guide for Scientists

    Hi, I have problems with the proof for task a I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
  24. chwala

    Prove that ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##

    I let, ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}## ##\tan^{-1}\left[\dfrac{1}{5}\right]- \dfrac{1}{4}\tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{16}## Then i let, ##\tan^{-1}\left[\dfrac{1}{5}\right] = α ...
  25. F

    B Have I proved some part of Fermat's last theorem?

    Have I proved Fermat last theorem? X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z. x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat...
  26. L

    How do you prove that ln(a^x) = xln(a) and a^x = e^xln(a) without using exponent rules?

    In the book "Calculus by Michael Spivak" it says that a^x = e^xln(a) is a definition. And I am not convinced to accept this as true without a proof.
  27. chwala

    Prove that the given inverse trigonometry equation is correct

    Ok in my approach i have, ##2 \tan^{-1} \left(\dfrac{1}{5}\right)= \sin^{-1} \left(\dfrac{3}{5}\right) - \cos^{-1} \left(\dfrac{63}{65}\right)##Consider the rhs, Let ##\sin^{-1} \left(\dfrac{3}{5}\right)= m## then ##\tan m =\dfrac{3}{4}## also let ##\cos^{-1} \left(\dfrac{63}{65}\right)=...
  28. porton

    Check my P=NP proof for errors (based on incompleteness of ZFC)

    Please check for errors my proof of P=NP: PDF file It is based on set theory and logic (incompleteness of ZFC). It uses also inversions of bijections, algorithms as arguments of other algorithms, reduction of SAT to another NP problem. [Moderator's note: link removed.]
  29. chwala

    Understanding the given proof of integers - Ring theory

    My interest is on the highlighted part ... Now to my question, in what cases do we have ##mn<(m,n)[m,n]?## I was able to use my example say, Let ##m=24## and ##n=30## for example, then ##[m,n]=120## and ##(m,n)=6## in this case we can verify that, ##720=6⋅120## implying that, ##mn≤...
  30. S

    I Multivariable fundamental calculus theorem in Wald

    i want to prove that if ##F:\mathbb{R}^n\to\mathbb{R}## is a differentiable function, then $$F(x)=F(a)+\sum_{i=1}^n(x^i-a^i)H_i(x)$$ where ##H_i(a)=\frac{\partial F}{\partial x^i}\bigg|_{x=a}##. the hint is that with the 1-dimensional case, convert the integral into one with limits from ##0## to...
  31. chwala

    I Proof of vector property in space

    My interest is on the associative property; is there anything wrong of showing and concluding proof by; ##c(\vec u⋅\vec v)=(c⋅\vec v)⋅\vec u.## or are we restricted in the prose?
  32. S

    I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'

    My question is motivated by the proof of TH 5.13 on p 84 in the 2nd edition of Linear Algebra Done Right. (This proof differs from that in the 4th ed - online at: https://linear.axler.net/index.html chapter 5 ) In the proof we arrive at the following situation: ##T## is a linear operator on a...
  33. P

    I Proof regarding congruence relation

    Let ##\Lambda## be a lattice and ##a, b \in \mathbb{R}^n##, then $$a \equiv b \text{ mod } \Lambda \Leftrightarrow a- b \in \Lambda$$ I want to prove the statement. For the left to right direction I would say, ##a \equiv b \text{ mod } \Lambda \Leftarrow a = b +k\Lambda##, where ##k \in...
  34. Feynstein100

    B Proof that pattern recognition is unending?

    So I've thought of an admittedly crude proof that the process of pattern recognition i.e. finding patterns, be they linguistic, mathematical, artistic, whatever, is a process that can never end. It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a...
  35. H

    A About Proof of Engel's Theorem

    in the Proof of Engel's Theorem. (3.3), p. 13: please, how we get this step: ##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##. Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent. Thanks in advance,
  36. hjam24

    I Write probability in terms of shape parameters of beta distribution

    Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$ If we know that A wins, his score is specified by B's score; he...
  37. Shreya

    The Definition of Torque - a proof

    I have been trying to understand this proof from the book 'Introduction to classical mechanics' by David Morin. This proof comes up in the first chapter of statics and is a proof for the definition of torque. I don't understand why the assumption taken in the beginning of the proof is...
  38. C

    Proof of ##M^n## (matrix multiplication problem)

    For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!
  39. mattTch

    I Proof of Column Extraction Theorem for Finding a Basis for Col(A)

    Theorem: The columns of A which correspond to leading ones in the reduced row echelon form of A form a basis for Col(A). Moreover, dimCol(A)=rank(A).
  40. C

    I Is an algorithm for a proof required to halt?

    I know that when giving an algorithm to prove something we need to prove two things about the algorithm ( there’s another option which is to show time-complexity but that’s optional since it’s irrelevant to the proof): 1. Correctness 2. That it halts But there are also algorithms/procedures...
  41. Vanilla Gorilla

    B Attempted proof of the Contracted Bianchi Identity

    My Attempted Proof ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## ##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}## So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ## Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0## Sum w/ inverse metric tensor twice ##g^{bn} g^{am}...
  42. VX10

    I A question about Young's inequality and complex numbers

    Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
  43. C

    Proof of angle in path difference formula for two slits

    For this I am trying to prove that angle theta between PQ and QO is equal to theta highlighted so that I know I can use theta is the path difference formula. I assume that the rays ##r_1## and ##r_2## are parallel since ##L >> d## My proof gives that the two thetas are equal, however I am...
  44. C

    Power Rule Proof: Get Help with Line 3 to Line 4

    For this proof, I am unsure how they got from line 3 to line 4. If I simplify and collect like terms for line 3 I get ##f'(a) = 4a^{n-1}## Would some please be able to help? Many thanks!
  45. V

    Logical Proof: Theorem (Truths of Logic) A iff ~~A

    My thought was to break up the sentence into its equivalent form: (A ->~~A) & (~~A -> A) From there I assumed the premise of both sides to use indirect proofs, so: 1. ~(A -> ~~A) AP 2. ~(~A or ~~A) 1 Implication 3. ~~A & ~~~A 2...
  46. E

    I Proof of Lorentz Gauge Existence: Help Understanding Schutz 8.3

    In Schutz 8.3, while proving that a Lorentz gauge exists, it is stated that $$\bar h^{(new)}_{\mu\nu} = \bar h^{(old)}_{\mu\nu} - \xi_{\mu,\nu} - \xi_{\nu,\mu} + \eta_{\mu\nu}\xi^\alpha_{,\alpha}$$ where ##\bar h## is the trace reverse and ##\xi^\alpha## are the gauge functions. Then it follows...
  47. J

    I Show ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##

    I would wish to receive verification for my proof that ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##. • It is easy to verify that ##A = \{a \in \mathbb{Q}: a^2 \leq 3\} \neq \varnothing##. For instance, ##1 \in \mathbb{Q}, 1^2 \leq 3## whence ##1 \in A##. • We claim that ##\sqrt{3}## is an...
  48. H

    I Lars Olsen proof of Darboux's Intermediate Value Theorem for Derivatives

    Here is Lars Olsen's proof. I'm having difficulty in understanding why ##y## will lie between ##f_a (a)## and ##f_a(b)##. Initially, we assumed that ##f'(a) \lt y \lt f'(b)##, but ##f_a(b)## doesn't equal to ##f'(b)##.
  49. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  50. chwala

    Show the proof by induction in the given problem

    My interest is solely on the highlighted part in red...hmmmmmmm :cool: taken a bit of my time to figure that out...but i got it. Looking for any other way of looking at it; I just realised that the next term would be given by; ##\dfrac{1}{4}(k+1)^2(k+2)^2-\dfrac{1}{4}k^2(k+1)^2##...
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