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. P

    MHB Trouble with understanding section of FOL completeness proof

    The Completeness Proof for First-Order Predicate Logic depends on if $\Phi$ is a set of consistent $\mathcal L$-formulas, then $\Phi$ is satisfiable. How is that constructed? There are a large number of Lemmas working from Machover's text Set theory, Logic and Their Limitations but I'm having...
  2. C

    Induction proof verification ##2^{n+2} < (n+1)## for all n ##\geq 6##

    $2^{n+2} < (n+1)!$ for all n $\geq 6$ Step 1: For n = 6, $256 < 5040$. We assume $2^{k+2} < (k+1)!$ Induction step: $2 * 2^{k+2} < 2*(k+1)!$ By noting $2*(k+1)! < (k+2)!$ Then $2^{k+3} < (k+2)!$
  3. Ineedhelp0

    I Parseval's theorem and Fourier Transform proof

    Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...
  4. G

    MHB Analytic geometry proof with triangle.

    Point D divides side AC, of triangle ABC, so that |AD|: |DC| = 1:2. Prove that vectors \vec{BD} = 2/3 \vec{BA} + 1/3 \vec{BC}.
  5. K

    Proof of mathematical theorems

    My question is simple. Can one prove any theorem in mathematics by having only a pen and a paper, or a super-computer for that matter? Since math is essentially all about theorems, and we usually take them as true. I guess someone went in and proved them at some point in our history. But some...
  6. A

    B What is the role of Thales' Theorem in the proof of Brahmagupta's Theorem?

    Brahmagupta's theorem: A cyclic quadrilateral is orthodiagonal (diagonals are perpendicular) if and only if the perpendicular to a side from the point of intersection of the diagonals bisects the opposite side. But I don't understand the first step of the proof for the necessary condition...
  7. Math Amateur

    MHB Proof that Arcsin x is continuous ....

    Can someone please help me to prove that the function f(x) = Arcsin x is continuous on the interval [-1, 1] ... Peter
  8. H

    I Proof using Rule of Disjunctive Amplification

    Book shows a proof where a conclusion is reached of: ##\neg r##. The next step says ##\neg r \lor \neg s## using the rule of disjunctive amplification. The rule of disjunctive amplification as I know it is ##p \implies p \lor q##. I don't see how from this you can also say ##\neg p \implies...
  9. PeroK

    I Fundamental Theorem of Algebra: Proof

    I found this video showing an elementary proof of the FTA.
  10. J

    Levi-Civita Identity Proof Help (εijk εijl = 2δkl)

    I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with: δjj δkl - δjl δkj = δkl - δlk Clearly I did not take the right approach in this proof and have no clue as to how to proceed.
  11. A

    MHB Proof that S (the successor function) is, in fact a Function.

    In axioms containg S one invariably finds: Sx = Sy -----> x = y The converse, which characterizes S as a function: x = y ------> Sx = Sy Is never shown. Neither is it shown as an Axiom of FOL or formal Theory of Arithmetic. From the basic axioms and rules of FOL, how does one go about...
  12. S

    I Proof of ##F## is an orthogonal projection if and only if symmetric

    The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...
  13. C

    What is the proof for Bertrand's theorem in celestial mechanics?

  14. J

    I Need help with a proof involving points on a quadratic

    Summary: Given three points on a positive definite quadratic line, I need to prove that the middle point is never higher than at least one of the other two. I am struggling to write a proof down for something. It's obvious when looking at it graphically, but I don't know how to write the...
  15. K

    Help with an epsilon-delta proof

    I have been struggling with this problem and also my friends. We are not the best at epsilon-delta proof and we have not found an understandable solution to this problem.
  16. E

    Comp Sci Time Complexity Algorithm Proof

    Use the formal definition of Big-Oh to prove that if f (n) and g(n) are nonnegative functions such that f (n) = O(g(n)), f (n) + g(n) = Ω(g(n)). By the definition of Big-Oh: If f(n) and g(n) are non-negative functions such that f(n) = O(g(n)) there must be positive constants c and n0 such...
  17. AutGuy98

    MHB Proof of an Infimum Being Equal to the Negative Form of a Supremum ()

    Hey guys, I'm kind of in a rush because I'll have to go to my classes soon here at USF Tampa, but I had one last problem for Intermediate Analysis that needs assistance. Thank you in advance to anyone providing it. Question being asked: "Let $A$ be a nonempty set of real numbers which is...
  18. AutGuy98

    MHB Proof of the Equality of Supremums (Or Something Like That Anyway :) )

    Hey guys, I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says: "Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so...
  19. Z

    Comp Sci Induction Proof for all strings: Can't understand the Question

    Hi, Can some body please explain me the following question: Use induction on ##n## to show that ##|t^n| = n |t| ## for all strings ##t## and all ##n## . Any idea how to that. I know we have a base case and an induction case but what would be the base case and what would be the induction case...
  20. E

    Proof with recursion and logarithms

    Homework Statement: Suppose f(n) is a function that accepts an integer n as a parameter. When called with n = 1, it executes 2 instructions. When called with a larger value of n, it executes 10n + 30 instructions, two of which are f(n/2). Prove that f(n) executes 10n lg n + 32n − 30...
  21. Iyan Makusa

    Comp Sci Proofing Big-O Notation: O(max(f(n), g(n)))

    So I know the formal definition of Big-O, which states that ##f(n) = O(g(n))## if and only if there exists ##{C > 0, n_0 > 0}## such that ##|f(n)|\leq{C}{g(n)}~\forall{n>n_0}##. Here's what I think the proof should go (please bear with me, I have no idea what I'm doing): Suppose there exists a...
  22. X

    Analytical proof that L and C are linear devices?

    I mean I know they are linear since they obey the ohms law. But I don't quite understand the reasoning that since, say, V=Ldi/dt and taking a derivative is a linear operation therefore it is a linear device?? I can verify that sin'(x) = cos(x) or sin(x+90) so the signal is time shifted but...
  23. H

    I Inadequate proof of Bloch's theorem?

    Suppose a wave function is a linear combination of 2 stationary states: ##\psi(x)=c_1\psi_1(x)+c_2\psi_2(x)##. By [5.52] and [5.53], we have ##\psi(x+a)=e^{iK_1a}c_1\psi_1(x)+e^{iK_2a}c_2\psi_2(x)##. But to prove [5.49], we need ##K_1=K_2##. That means all the eigenvalues of the "displacement"...
  24. kmm

    I Bernard Schutz Proves Invariance of Interval

    I've been going through Bernard Schutz's A First Course in General Relativity, and I'm hung up on his "proof" of the invariance of the interval. At the beginning of section 1.6, he claims that he will prove the invariance of the interval, and after a few lines shows that the universality of the...
  25. Manasan3010

    Unable to prove this Equation

    The attempt at a solution ## {\log_{2021}(tan\ 1^\circ)} +{\log_{2021}(tan\ 2^\circ)} +{\log_{2021}(tan\ 3^\circ)} +...+{\log_{2021}(tan\ 89^\circ)} = 0 \\ Antilog_{2021}[{\log_{2021}(tan\ 1^\circ)} +{\log_{2021}(tan\ 2^\circ)} +{\log_{2021}(tan\ 3^\circ)} +...+{\log_{2021}(tan\ 89^\circ)} ]...
  26. Y

    MHB Cartesian Product - Proof

    Dear all, I am trying to prove a simple thing, that if AxA = BxB then A=B. The intuition is clear to me. If a pair (x,y) belongs to AxA it means that x is in A and y is in A. If a pair (x,y) belongs to BxB it means that x is in B and y is in B. If the sets of all pairs are equal, it means...
  27. F

    I What is the proof of the rules of significant figures?

    Please prove the rules of significant figures. I do not know why when multiplying and dividing we have to retain the same number of significant figures as in the number with the least of them.
  28. I

    Epsilon delta proof of the square root function

    Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...
  29. matqkks

    I Proof of the Division Algorithm

    In many books on number theory they define the well ordering principle (WOP) as: Every non- empty subset of positive integers has a least element. Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it...
  30. matqkks

    MHB Proof of the Division Algorithm

    In many books on number theory they define the well ordering principle (WOP) as: Every non- empty subset of positive integers has a least element. Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible...
  31. B

    MHB TFAE proof involving limit and convergent sequence

    Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the following statements are equivalent: i) limx→∞ f(x) = L ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. Not even sure how to begin this one, other than the fact...
  32. P

    Kepler's laws and proof using angular momentum

    a) Kepler's first law states that a planet like Earth displays an elliptical orbit with the sun in focus. Using M = dL/dt, prove that a planet cannot leave its plane of orbit. Note: M here is an externally applied torque that the sun exerts on the planet. diagram of the situation described b)...
  33. R

    B A proof of the fundamental theorem of calculus

    is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it? i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it. i know...
  34. M

    Proof a property for a 3x3 matrix

    Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
  35. Adgorn

    I Attempting to find an intuitive proof of the substitution formula

    Hello everyone. First off, I'm sorry if this post is excessively long, but after tackling this for so many hours I've decided I could use some help, and I need to show everything I did to express exactly what I wish to do. Also, to be clear, this post deals with integration by substitution. Now...
  36. A

    MHB Help with proof of equivalence

    Consider the equivalence: (∀v Fv -> p) <=> (∃u Fu -> p) Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v. Can someone please offer a proof of such equivalence. Many thanks. am
  37. fresh_42

    Insights How to Write a Math Proof and Their Structure

    Continue reading...
  38. ali PMPAINT

    A geometric proof (minimizing the length of two lines in a rectangle)

    So, I know it can be proven using calculus, but I need the geometric one. So, I got that ^c=^d and therefor, the amount of increment in one of a, is equal to the other(^e=^b). (Also 0<a+b<Pi/2) And AP'=BP'=BD/sin(a) and BP=BD/sin(a+b) and AP=BD/sin(a-b). AP'+BP'=2AP'=2BD/sin(a) and...
  39. S

    Prove the decomposition of a graph w/ even edges produce a 2-path set

    For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2. The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two...
  40. F

    I Proof of Addition Reversibility

    How can you prove that ##f(x)=g(x) \Leftrightarrow f(x)+C=g(x)+C##
  41. T

    I Question about Cantor's Diagonal Proof

    Dear friends, I was wondering if someone can explain how Cantors diagonal proof works. This is my problem with it. He says that through it he finds members of an infinite set that are not in another. However, 2 and 4 are not odd numbers, but all the odd numbers equal all the whole numbers. If...
  42. S

    Proving |a|=|-a|: Using Cases and Triangle Inequality"

    Problem Statement: Prove that |a|=|-a| Relevant Equations: ##|a|= a, ## if ## a \geq 0 ## and -a, if ## a \leq 0 ## Somewhat stumped on where to start... i know that we need to use cases. If we consider ##a\geq 0##, then are we allowed to use the fact that ##|-a|=|-1|\cdot|a| = |a| ##? This...
  43. S

    Help with a proof (Spivak Ch. 1, 1,iii

    Note sure if this belongs in the Basic Math category or Calc & Beyond section. I want to make sure I am on the right track here. Here is what i have so far: x^2 = y^2 Multiply both sides by x^-1 twice (invoking P7) x^2 \cdot x^{-1} = y^2 \cdot x^{-1} x \cdot x^{-1} = y^2 \cdot x^{-1}...
  44. Math Amateur

    MHB Proof of the Cauchy-Schwarz Iequality .... Garling, Proposition 11.3.1 .... ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help to fully understand the proof of...
  45. U

    I Missing(?) rigor in proof involving countable union of countable sets

    My question concerns the portion of the proof stating, “...we set up a correspondence between the elements of U(A_n), for n in N, and a subset of S by making the element a correspond to (m, n) if A_m is the first set in which a appears, and a is the nth element of A_m.” In particular, I am...
  46. U

    I Error(?) in proof that the rational numbers are denumerable

    If someone can straighten out my logic or concur with the presence of a mistake in the proof (even though the conclusion is correct, of course), I would be much obliged. I’m looking at the proof of the corollary near the middle of the page (image of page attached below). I simply don’t find...
  47. Arman777

    Proof that Variation of Integral is Equal to Integral of the Variation

    I actually don't know how to proceed. I tried something like this The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$ where ##f'(x)=F(x)## However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$ where ##f'(x)=F(x)##. For the...
  48. Dfpolis

    A How does Bell's 1964 theorem address detector independence and local realism?

    I have some questions about J. S. Bell’s famous theorem as presented in his1964 paper.1 These are about his theoretical assumptions and reasoning, not about experimental observations such as Aspect-type experiments. While some questions relate to the experiments, others do not because Aspect’s...
  49. M

    How can the inequality cosx ≥ (1-x^2/2) be proven?

    How can the inequality ##cosx \ge(1-x^2/2)## be proved? Would you please explain how to prove this inequality? This is the only equation that I could think of. ##1\ge cosx \ge 0## but I cannot use it here. Source: Thomas's Calculus, this is from an integration question there. Thank you.
  50. mertcan

    I Bayesian Information Criterion Formula Proof

    Hi everyone, while I was digging arima model I saw that BIC value is given as $k*log(n)-2*log(L)$ where $L$ is the maximized value of likelihood function whereas $k$ is number of parameters. I have found the proof of AIC but no any clue about that. I wonder how it is derived. Could you help me...
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