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. 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...
  2. 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...
  3. 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.
  4. 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...
  5. 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...
  6. 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...
  7. 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}##...
  8. 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...
  9. 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...
  10. 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...
  11. S

    B Did President Garfield really come up with an alternate proof?

    I'm talking about the Pythagorean Theorem, which seems to have an alternate proof attested to him! http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html
  12. L

    MHB Question about proof of the linear independence of a dual basis

    This is from Kreyszig's Introductory Functional Analysis Theorem 2.9-1. Let $X$ be an n-dimensional vector space and $E=\{e_1, \cdots, e_n \}$ a basis for $X$. Then $F = \{f_1, \cdots, f_n\}$ given by (6) is a basis for the algebraic dual $X^*$ of $X$, and $\text{dim}X^* = \text{dim}X=n$...
  13. Stoney Pete

    I An easy proof of Gödel's first incompleteness theorem?

    Hi everybody, Do you think the following reconstruction of Gödel's first incompleteness theorem is basically correct, or at least in the right ballpark? In my view, this incompleteness result basically turns on the mismatch between the indenumerability of the powerset of ℕ and the enumerability...
  14. Eclair_de_XII

    Proof for convergent sequences, limits, and closed sets?

    Homework Statement "Let ##E \subset ℝ##. Prove that ##E## is closed if for each ##x_0##, there exists a sequence of ##x_n \in E## that converges to ##x_0##, it is true that ##x_0\in E##. In other words, prove that ##E## is closed if it contains every limit of sequences for each of its...
  15. RoboNerd

    Finding GCD with Fibonacci: Base Case

    Homework Statement Suppose that m divisions are required to find gcd(a,b). Prove by induction that for m >= 1, a >= F(m+2) and b>= F(m+1) where F(n) is the Fibonacci sequence. Hint: to find gcd(a,b), after the first division the algorithm computes gcd(b,r). Homework Equations Fibonacci...
  16. O

    Step potential, continuous wave function proof

    Homework Statement I am being asked to show that the wave function ψ and dψ/dx are continuous at every point of discontinuity for a step potential. I am asked to make use of the Heaviside step function in my proof, and to prove this explicitly and in detail. Homework Equations...
  17. C

    MHB Which fractions can you make (with proof)?

    Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/(2x). Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?
  18. V

    B Fermat's Last Theorem; unacceptable proof, why?

    Wikipedia says Fermat's last theorem has the greatest number of failed proofs in history. I presume this simple "proof" is one of them. It must have been thought up before me. I first considered it years ago when I first heard of the problem. Figured it was so simple someone else must have...
  19. N

    An alternative proof (Hopefully not an alternative fact)

    Homework Statement Hi all, I'm currently studying the amazing Calculus by Spivak. Whenever I study textbooks I always attempt to do all the examples and proofs in the text before looking at the answers. (Whether this is a good thing or a bad thing I don't know, the examples are similar to the...
  20. Math Amateur

    MHB Proof of Bolzano-Weierstrass on R .... .... D&K Theorem 1.6.2 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.6.2 ... Duistermaat and Kolk"s Theorem 1.6.2 and its proof read as follows:In the...
  21. L

    Proof of uniqueness of limits for a sequence of real numbers

    Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...
  22. Euler2718

    Can Three Non-Collinear Points Always Define a Projective Plane?

    Homework Statement Let P(W) be a projective space whose dimension is greater than or equal to 2 and let three non-colinear projective points, [v_{1}],[v_{2}],[v_{3}]\in P(W) . Prove that there is a projective plane in P(W) containing all three points. Homework EquationsThe Attempt at a...
  23. Mr Davis 97

    Proof by Contradiction: Showing a ≤ b when a ≤ b1 for every b1 > b

    Homework Statement Let ##a,b \in \mathbb{R}##. Show if ##a \le b_1## for every ##b_1 > b##, then ##a \le b##. Homework EquationsThe Attempt at a Solution We will proceed by contradiction. Suppose that ##a \le b_1## for every ##b_1 > b##, and ##a > b##. Let ##b_1 = \frac{a+b}{2}##. We see that...
  24. C

    A Is the proof of these results correct?

    Hello, Below are two results with their proof. Of course, there may be several ways to prove these results, but I just need some checking. Can someone check carefully if the math is OK ? (but very carefully, because if there is a failure, I will be murdered :-) ) ? thx. Claim 1: Let ##L/K## be...
  25. J

    MHB Congruence Class Proofs: Tips and Examples

    Hi, I have tried the question as attached. I am not sure if I am correct. You help is greatly appreciated. Thanks in advance!
  26. R

    Induction Proof for A^n = 1 2^nProve your formula by mathematical induction.

    <Moderator's note: Moved from a technical forum and thus no template.> A^n = 1 2^n 0 1 Prove your formula by mathematical induction. I began by taking A to successive powers but not sure of what my formula should be. A^0 = 1 0 , A^1 = 1 2 , A^2 = 1 4 , A^3 = 1 6 ...
  27. alan123hk

    B Proof of the relation between antenna aperture and gain

    Where can I find strict mathematical proof of the relation between antenna aperture and gain which is applicable to any type of antenna ? Aeff = Gain * (lambda^2) / (4*Pi) Aeff - Antenna Effective Aperture Gain - Antenna Gain lamdda - wavelength Pi - 3.14159 Many textbooks just show the...
  28. A

    Skew bending in a circular cross section (proof)

    Good day all I'm looking for the proof of stress generated in case of skew bending applied in acircular cross section ( I browsed internet the whole day without finding anything convincing) we use with many thanks in advance!
  29. F

    Limit proof as x approaches infinity

    Homework Statement Verify the following assertions: a) ##x^2 + \sqrt{x} = O(x^2)## 2. Homework Equations If the limit as x approaches ##\infty## of ##\frac {f(x)}{g(x)}## exists (and is finite), then ##f(x) = O(g(x))##. The Attempt at a Solution Let ##\epsilon > 0##. We solve for ##\delta##...
  30. J

    MHB Troubleshooting Euclid's Lemma Proof in Modular Arithmetic

    Encountered difficulties in proving the attached image. Greatly appreciate for the help!
  31. Let'sthink

    I Proof without words for Heron's formula

    I was thinking about a diagram (in the category of proof without words) for Hero's formula for area of a triangle with sides a, b, and c and given that 2s = a+b+c. A = √[(s(s-a)(s-b)(s-c)] I tried to develop one but could not. Can anybody give or give me an hint to proceed.
  32. Eclair_de_XII

    Can anyone check my proof involving least-upper-bounds?

    Homework Statement "If ##x=sup(S)##, show that for each ##\epsilon > 0##, there exists ##a∈S## such that ##x-\epsilon < a ≤ x##" Homework Equations ##x=sup(S)## would denote the least upper bound for ##S## The Attempt at a Solution "First, we consider the case where ##x=sup(S)∈S##. Then...
  33. bubblescript

    Is Proof of A Subset of Union of Family Valid?

    Homework Statement If ##\mathcal{F}## is a family of sets and ##A \in \mathcal{F}##, then ##A \subseteq \cup \mathcal{F}##. Homework Equations ##A \subseteq \cup \mathcal{F}## is equivalent to ##\forall x(x \in A \rightarrow \exists B(B \in \mathcal{F} \rightarrow x \in B))##. The Attempt at...
  34. W

    Proof of Wheatstone bridge equation

    Homework Statement Prove the following equation: ## \Delta U=\frac {R_1R_4}{(R_1+R_4)^2}(\frac {\Delta R_1}{R_1}-\frac {\Delta R_2}{R_2}+\frac{\Delta R_3}{R_3}-\frac{\Delta R_4}{R_4})E## This is used in Wheatstone bridge Homework Equations [/B] U=RI The Attempt at a Solution This has...
  35. tomwilliam2

    I What is the Proof of an Inequality for Three Positive Numbers?

    I'm trying to do some practice Putnam questions, and I'm stuck on the following: For ##a,b,c \geq 0##, prove that ##(a+b)(b+c)(c+a) \geq 8abc## (https://www.math.nyu.edu/~bellova/putnam/putnam09_6.pdf) I started off by expanding the brackets and doing some algebraic rearranging, but I don't...
  36. shihab-kol

    B Prove Limit Rule: Learn the Constant Concept

    Hello, I would like to begin by saying that this does not fall into any homework or course work for me. It is just my interest. I need to prove that limit of a constant gives the constant it self. Can some one provide a link? I have exams or I would have searched myself but unfortunately I don't...
  37. M

    I Proof that Galilean & Lorentz Ts form a group

    The Galilean transformations are simple. x'=x-vt y'=y z'=z t'=t. Then why is there so much jargon and complication involved in proving that Galilean transformations satisfy the four group properties (Closure, Associative, Identity, Inverse)? Why talk of 10 generators? Why talk of rotation as...
  38. S

    Find Real Values of k for Purely Real ##u##

    Homework Statement Find all possible values of ##k## that make ##u = \frac{k+4i}{1+ki}## a purely real number. Homework EquationsThe Attempt at a Solution I calculated the complex conjugate which was ##\frac{5k}{k^2+1} + \frac{4-k^2}{k^2+1}i##. So to prove this do I just solve...
  39. F

    Proof about cycle with odd length

    Homework Statement In the following problems let ##\alpha## be a cycle of length ##s##, and say ##\alpha = (a_1a_2 . . . a_s)##. 5) If ##s## is odd, ##\alpha## is the square of some cycle of length s. (Find it. Hint: Show ##\alpha = \alpha^{s+1}##) Homework EquationsThe Attempt at a Solution...
  40. Math401

    I Proof: 0.9999 does not equal 1

    Or rather counter proof. They said x=0.999... 10x=9.999... 9x=9.999...-x 9x=9 x=1 but this is obviously wrong, you can't substract infinity from infinity unless you consider infinity a number and if so then you would get 8.99...1 and not 9. either way 0.999...= 1 is wrong. and is not different...
  41. D

    I Proof that 0 + 0 +....+ 0 +.... = 0

    Is there any rigorous way of proving this? I tried using geometric series of ever diminishing ratio and noticing that 0 is always less than each term of the series, then 0 + 0 +...+ 0 +... must be always less than ## \frac{1 } {1-r} - 1 ##. (*) Eventually, as r goes to 0 so does ## \frac{1 }...
  42. J

    MHB Prime Or Composite - Proof required?

    n^2 - 14n + 40, is this quadratic composite or prime - when n ≤ 0. Determine, all integer values of 'n' - for which n^2 - 14n + 40 is prime? Proof Required. ps. I can do the workings, but the 'proof' is the problem. Many Thanks John.
  43. J

    MHB Proof by Induction - in Sequences.

    Dear ALL, My last Question of the Day? Let b1 and b2 be a sequence of numbers defined by: b_{n}=b_{n-1}+2b_{n-2} where $b_1=1,\,b_2=5$ and $n\ge3$ a) Write out the 1st 10 terms. b) Using strong Induction, show that: b_n=2^n+(-1)^n Many Thanks John C.
  44. G

    How Does the Graph Laplacian Explain the Multiplicity of Eigenvalue Zero?

    Homework Statement Let ##G## be a non-directed graph with non negative weights. Prove that the multiplicity of the eigenvalue ##0## of ##L_s## is the same as the number of convex components ##A_1,\dots, A_k## of the graph. And the subspace associated to the eigenvalue ##0## is generated by the...
  45. J

    MHB Proof & Structures: Showing n≤0 for Prime/Composite Number

    Hi There, My apologies, there was an error...in a previous question, which I POSTED ....last week. This question has now been withdrawn, & replaced with the following : ----------------------------------------------------------------------------------------------------------------- a) Show...
  46. D

    LaTeX Modeling 3 species using omnivory models. Latex code is included for the proof of continuity, etc

    \chapter{Sensitivity Analysis} The first step in our method to obtain the sensitivity of each parameter value is to differentiate the right hand side of each model with respect to each model parameter. The partial derivatives for the right hand side of our linear response model...
  47. Van Ladmon

    A proof about tensor invariants

    Homework Statement How to proof the following property of tensor invariants? Where: ##[\mathbf{a\; b\; c}]=\mathbf{a\cdot (b\times c)} ##, ##\mathbf{T} ##is a second order tensor, ##\mathfrak{J}_{1}^{T}##is its first invariant, ##\mathbf{u, v, w}## are vectors. Homework Equations...
  48. J

    MHB Proof & Structure: Solve (¬ p V q) ↔ ( p Λ ¬ q) - John

    Dear ALL, Today, I am really struggling to complete...an important Assignment on time? In particular, this Question has ...Frazzled me, re Truth Tables etc etc...? Any good advice, by close of business - greatly appreciated...
  49. J

    MHB Is the Converse of the Given Statement True for Any Positive Integer n?

    if n is a positive integer greater than 2 and m the smallest integer greater than or = n, that is a perfect square. Let a = m-n. Show that if n is prime, then a is not a perfect square. Also, is the converse of above true, for any integer n? any guidance, will be much appreciated? Thanks
  50. Rodrigo Schmidt

    I Doubt about proof on self-adjoint operators.

    So the statement which the proof's about is: For every linear transformation ##A##(between finite dimension spaces), the product ##A^*A## is self-adjoint. So, the proof is: ##(A^*A)^*=A^*A^{**}=A^*A## What i don't understand is why ##(A^*A)^*=A^*A^{**}##. Isn't that true only if ##A## and...
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