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. 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 ...
  2. 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...
  3. 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!
  4. 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##...
  5. J

    MHB Troubleshooting Euclid's Lemma Proof in Modular Arithmetic

    Encountered difficulties in proving the attached image. Greatly appreciate for the help!
  6. 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.
  7. 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...
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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 }...
  17. 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.
  18. 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.
  19. 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...
  20. 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...
  21. D

    LaTeX Why Analyze Sensitivity in Omnivory Models?

    \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...
  22. Van Ladmon

    How to Prove the Property of 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...
  23. 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...
  24. 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
  25. 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...
  26. C

    MHB Question 29- How to conclude the following proof.

    Dear Everyone, I have trouble writing the conclusion of the proof. 29. The square of every odd integer is one more than an integral multiple of 4. Work: Let $n\in\Bbb{Z}$ If n is odd, then $n^2=1+4k$ for some $k\in\Bbb{Z}$. Examples Let n=3. Then k=2. Let n=5. Then k=6. Let n=21. Then k=110...
  27. M

    Why Assume Limit of \( S_{n-1} = S_n \)?

    Homework Statement If the sum of a sub n to infinity (n=1) converges then the limit of n as n tends to infinity of an = 0 Homework EquationsThe Attempt at a Solution an =(a1+a2+...an)-(a1+...+an-1) = limit of an (n tends to infinity) = sn -s(n-1) =0 The area I'm confused is why do we assume...
  28. A

    Why Is the Angle 90 Degrees in Elastic Collisions of Equal Mass?

    Homework Statement Prove that in the elastic collision of two objects of identical mass, with one being a target initially at rest, the angle between their final velocity vectors is always 90 degrees. Homework Equations m1v1+m2v2 = m1v1'+m2v2' 1/2m1v1^2 +1/2m2v2^2 = 1/2m1v1'^2 + 1/2m2v2'^2...
  29. Euler2718

    Proof of sequence convergence via the "ε-N" definition

    Homework Statement Prove that \lim \frac{n+100}{n^{2}+1} = 0 Homework Equations (x_{n}) converges to L if \forall \hspace{0.2cm} \epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N\in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall n\geq N \hspace{0.2cm} , |x_{n}-L|<...
  30. R

    MHB Stuck on a trigonometric identity proof....

    $\frac{1 -\cos A}{1 + \cos A} = (\cot A - \csc A)^2$
  31. G

    Understanding the Proof: Why Does It Work?

    Homework Statement 2. Questions I want to know how this proof works, but there are several things I don't understand. (1) How does the ##k\epsilon## appears? ##k## is the lipschitz constant, but what about ##\epsilon##? Is the author taking ##||x-y||<\epsilon##? (2) What does the...
  32. B

    I Where can I find a proof of the Swiss cheese theorem?

    Does anyone know where I can find a proof of this theorem? Theorem: The Euclidean space ##\mathbb{R}^2## is not the union of nondegenerate disjoints circles.
  33. Math Amateur

    MHB How does Conway's Theorem 2.29 Derive the Simplified Expression for f(z)?

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter III Elementary Properties and Examples of Analytic Functions ... Section 2: Analytic Functions ... ... I need help in fully understanding aspects of Theorem 2.29 ...
  34. F

    Proofing Bounds of Natural Numbers Set in Math: Is it Clear?

    Homework Statement Consider the sets below. For each one, decide whether the set is bounded above. If it is, give the supremum in ##\mathbb{R}##. Then decide whether or not the set is bounded below. If it is, give the infimum. Finally, decide whether or not the supremum is a maximum, and...
  35. PsychonautQQ

    A Proof of Seifert-Van Kampen Theorem

    To help me with this question, I think you'll need to have access to the proof, it's pretty involved and technical. I'm going the proof found in John M. Lee's "Introduction to topological manifolds", but I suspect that the proof will be the same no matter where you find it. Let ##U,V## be the...
  36. N

    MHB Major issue to handle this proof

    Good morning. My problem is as follow: I have an event assuming A. The probability that A occurs at time t is: p(t)= e^{-bt}*|\sin(at)|. Where a,b are positive parameters. We divide the time in small step times let's say \delta t= 0.125, Then, we count how many time A occur for $t \in [0...
  37. J

    Proof for Γ(p+1/2) using Double Factorial and nΓ(n)

    Homework Statement Prove that for a positive integer, p: https://www.physicsforums.com/posts/5859454/I've tried this to little avail for the better part of an hour - I know there's a double factorial somewhere down the line but I've been unable to expand for the correct expression in terms of...
  38. F

    Proof of Bounded Set without Max or Min: (0,2) in (0,2)

    Homework Statement Give an example of a bounded set that has neither a maximum nor a minimum. (The proof below is given by the book). We claim that the set ##(0,2)## is bounded and has neither a maximum nor a minimum. Proof: For each ##x \epsilon (0,2)##, we know that ##0 < x < 2##. Therefore...
  39. S

    Inertial Reference Frame Proof

    Consider a specific reference frame (0XYZ) attached to Earth. A point (origin) being selected, coordinates are ascribed along with a vector basis. This reference is non-inertial because it is locked to Earth and the acceleration of Earth is not zero. Suppose upon rising one morning I felt...
  40. PsychonautQQ

    I Proof that retract of Hausdorff space is closed

    I'm reading this proof on the matter: https://math.stackexchange.com/questions/278755/show-that-a-retract-of-a-hausdorff-space-is-closed How do we know that the final neighborhood they come up with is disjoint from A?
  41. Pushoam

    I DU = TdS - p dV is valid for irreversible process -- Proof

    The book says that all quantities in eq. 14.17 are state- functions. How do we know that? Earlier I read that both W and Q are not state functions. Now, since S is state-function, it means that tem. is not a state- function, isn't it? Does it mean that for a reversible process, both W and Q are...
  42. B

    Proof that Algebraic Integers Form a Subring

    Homework Statement The set ##\Bbb{A}## of all the algebraic integers is a subring of ##\Bbb{C}## Homework EquationsThe Attempt at a Solution Here is an excerpt from my book: "Suppose ##\alpha## an ##\beta## are algebraic integers; let ##\alpha## be the root of a monic ##f(x) \in...
  43. Z

    Proof of convex conjugate identity

    Homework Statement Prove that the conjugate of ##g(x) = f(Ax + b)## is ## g^*(y) = f^*(A^{-T}y) - b^TA^{-T}y ## where A is nonsingular nXm matrix in R, and b is in ##R^n##. Homework Equations This is from chapter 3 of Boyd's Convex Optimization. 1. The conjugate function is defined as ##...
  44. S

    MHB Proof of Divergence: (-1)^n Sequence

    Prove that the sequence :(-1)^n diverges by using the ε-definition of the limit of a sequence
  45. Mr Davis 97

    I Proving Convergence: Solving the Limit of 1/(6n^2+1) = 0

    I am trying to show that ##\displaystyle \lim \frac{1}{6n^2+1}=0##. First, we have to find an N such that, given an ##\epsilon > 0##, we have that ##\frac{1}{6n^2+1} < \epsilon##. But in finding such an N, I get the inequality ##n> \sqrt{\frac{1}{6}(\frac{1}{\epsilon}-1)}##. But clearly with...
  46. Moayd Shagaf

    I Advancing LQG: Challenges and Possibilities for Experimental Proof

    What issues we have in prove LQG experimentally? and It is eaiser to prove than string theory? If so , why There's a lot of string theorist than LQG?
  47. PsychonautQQ

    I How Does Closure in a Neighborhood Imply Membership in a Set?

    So I'm trying to understand a small part in the proof about how every 1-manifold is triangulable. Let G be contained in K and let x be a limit point of G. Let U be a neighborhood of K that intersects G in finitely many closed neighborhoods, thus U intersect G is closed in G and thus x is in G...
  48. J

    I How to Prove the Solid Angle Formula for Vector Fields?

    Hello, I would like to ask you for hints to proof this: \int^4 pi \omega \omega \,d\omega =%fraction{4}{3} \pi where omega is vector. Do you have any hints for me? (Not seeing for solution, just hints).
  49. Mr Davis 97

    I Cantor's decimal proof that (0,1) is uncountable

    Assume for contradiction that there does exist a function ##f : \mathbb{N} \rightarrow (0,1)## that is a bijection. ##\forall m \in \mathbb{N}##, ##f(m)## is a real number between 0 and 1, and we represent it using the decimal notation ##f(m) = .a_{m1}a_{m2}a_{m3}...##. We assume that every real...
  50. Mr Davis 97

    Proof that algebraic numbers are countable

    Homework Statement Fix ##n,m \in \mathbb{N}##. The set of polynomials of the form ##a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0## satisfying ##|a_n| + |a_{n-1}| + \cdots + |a_0| \le m## is finite because there are only a finite number of choices for each of the coefficients (given that they...
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