What is Mathematical proof: Definition and 61 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. 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...
  2. A

    Understanding Set Operations: A Closer Look at A ∖ (A ∖ B) = A ∩ B

    ##A ∖ B## can't include any elements that are not in ##A##, so it is the same as saying ##A∖(A∩B)##; it's exactly the elements of ##A## except those in ##A∩B##. ##A∖(A∖(A∩B))## is exactly the elements of ##A## except those in (exactly the elements of ##A## except those in ##A∩B##). This is the...
  3. C

    Purely Inductive Circuit -- Mathematical proof for current lag

    how we can mathematically prove that in a purely inductive circuit current lags behind voltage by a phase angle of π/2?
  4. CheesyPeeps

    Prove True/False that n^3-n is Always Divisible By 6

    Homework Statement For all natural numbers n, prove whether the following is true or false: n3-n is always divisible by 6. From SQA Advanced Higher Mathematics 2006 Exam Paper Homework Equations I can choose from the following types of proof: Direct proof Proof by contradiction Proof by...
  5. koustav

    Are Spacelike and Timelike Orthogonal: Mathematical Proof Explained

    are spacelike and timelike orthogonal?what is the mathematical proof
  6. T

    Using symbolic logic in mathematical proof?

    is this a practical way of proving math theorems? i asked because when i tried, it seemed difficult for me to decide as to how exactly i should translate theorems and given statements into logical forms and since there are so many different ways, i do not know which one is correct. For example...
  7. A

    Mathematical proof for oil drilling

    hi guys, I'm supposed to write a paper and do some research on all aspects of drilling regarding torque and force need to drill an oil well and everything from start to finish and to provide mathematical proof and calculations. I don't know where to start and what are the equations used to...
  8. Memocyl

    Mathematical Proof of Kepler's First Law of Orbits

    Hello friends (I hope :biggrin:), For a maths project I am working on, I need to be able to prove the equation for an elliptical orbit, related to Kepler's first law: and p = a(1-e2) (or should be as p can be replaced by that value) Where: r = distance from sun to any point on the orbit p =...
  9. D

    Mathematical proof (Drawing a help line)

    Homework Statement I'm doing quite a strict proof in school. Where we should proof something and use mathematical language and symbols. Homework Equations The Attempt at a Solution To proof what I have to proof I need to draw some help lines. As for instance the "red" one I did from A to B...
  10. TheMathNoob

    Proving a/b+b/a >= 2 using Mathematical Proof | Homework Help

    Homework Statement Let a,b be in the positive reals. Prove a/b+b/a is >=2 Homework EquationsThe Attempt at a Solution I have no idea. Maybe add the two ratios: (a^2+b^2)/a*b and then try to analyze separately the numerator and denominator?
  11. Patrick Sossoumihen

    Mathematical Proof: Does Euclidian Geometry Hold in Riemannian?

    Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?
  12. Comeback City

    Mathematical Proof that Photons have No Rest Mass

    Using these equations I am about to prove that photons have a rest mass of zero (mathematically) ________________________________________________________________________________________ E=hc/λ Photon Energy Equation E2=(pc+mc2)2 Mass-Energy Equivalence with Momentum Equation p=h/λ Momentum...
  13. sciencejournalist00

    Is this mathematical proof that beamsplitters entangle photons

    Prof. S Lakshmi Bala from Department of Physics, Madras, India writes a blackboard of equations which show how beamsplitters used alone affect the wavefunctions of input photons. It seems that it depends on the number of photons you use and in which input port to get you a different entangled...
  14. D

    Mathematical Proof That a 45 Degree Angle Launch is Best for

    Mathematical Proof That a 45 Degree Angle Launch is Best for Displacement how do i solve this
  15. J

    Mathematical proof of statement

    There is an article on Wikipedia gravity assist and in the explanation of the term, author gives an analogy: Isn't this violating the law of conservation of energy and momentum? Can someone make a mathematical proof of this? I tried, but the variables like the mass of the train and the tennis...
  16. D

    Proving uniqueness of a mathematical object

    As I understand it, the usual method for proving uniqueness of a mathematical object (for example the identity element of a group) is to use a proof by contradiction. Now, for example, if we have ##a## such that ##ax=b## and we want to prove this is unique, we start by assuming the contrary...
  17. A

    A proof related to 2 X 2 matrix

    Let A be a 2 X 2 matrix such that AX = XA for all 2 X 2 real matrices X. Show that A =kI for some k belonging to R
  18. K

    Are Projection Mappings considered Quotient Maps?

    The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...
  19. C

    Mathematical induction example

    Homework Statement A step in this process of proving Sn: 1+4+7+...+(3n-2) = n(3n-1)/2 confuses me. I hope someone can clarify this for me. I do not require the work done, I need clarification on a step only. Thanks! Homework Equations After assuming n=k, we say Sk: 1+4+7+...+(3k-2) =...
  20. M

    Orthogonality of Associated Laguerre Polynomial

    I have a problem when trying to proof orthogonality of associated Laguerre polynomial. I substitute Rodrigue's form of associated Laguerre polynomial : to mutual orthogonality equation : and set, first for and second for . But after some step, I get trouble with this stuff : I've...
  21. J

    Mathematical proof for non linear system

    Homework Statement Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time. Block diagram is the product of x1(t) and x2(t) going through a block step...
  22. Seydlitz

    How to do Mathematical proof for beginner

    Hello all, I know that perhaps this question has cropped frequently in a lot of places, but I would like to be quite specific in my case. I've just finished High School with UK A level syllabus. The syllabus inside Math A level is mostly straight solving equation type of questions, such as for...
  23. A

    Need Help with Mathematical Proof? - Introducing Myself & Seeking Assistance

    Dear all, I am new to this forum and this is my first post. I hope this is the right place to open this topic. I have a homework for my mathematical methods for physics course and i am kind of stuck into a question: v=ψ∇β show that ∇xv is perpendicular to v. Here v is a vector x is cross...
  24. A

    Mathematical proof for positive feedback

    Is there a mathematical proof that the positive feedback makes the op-amp saturated? http://i.imgur.com/71PNh.png
  25. T

    How useful is mathematical proof as a mechanical engineer?

    Specifically I plan on specializing as a Mechatronics engineer. I recently bought the book "Mathematical Proofs: A Transition to Advanced Mathematics" and I plan to study it on my own due to curiosity and interest towards mathematics. I would like to listen on one's opinion on how useful it...
  26. V

    Pairwise disjoint mathematical proof

    Set 1 {{1, 2}, {2, 3}, {1, 2, 3}, {2, 3, 4}, {2, 3, 5}, . . . , {2, 3, n}}; Set 2 {{3, 5}, {5, 4}, {3, 5, 4}, {4, 5, 1}, {4, 5, 2},{3, 5, 6}, {3, 5, 7}, . . . , {3, 5, n}}; Set 3 {{3, 1}, {1, 4}, {3, 1, 4}, {1, 4, 2}, {1, 3, 5}, {1, 3, 6}, . . . , {1, 3, n}}; Set 4 {{1, 5}, {5, 2}, {1...
  27. AnTiFreeze3

    Mathematical Proof That The Media is Sexist and Bad at Math

    "Mathematical Proof That The Media is Sexist and Bad at Math" Here's is a link to my favorite comedy website: Cracked.com (The website itself may be NSFW, but this particular article doesn't get vulgar). I thought that this was an interesting article that points out what is too obvious for the...
  28. S

    Element Selection for Voltage Doubler Design with Mathematical proof

    I need to design a voltage doubler that takes in 115V-50Hz a.c. and produces a doubled voltage output of 230V dc. I am using this circuit: I need help in the mathematics part: i) How do I find the optimum value of capacitance? (Using high capacitance values reduces ripples; but can...
  29. H

    Where Can I Find a Good Introductory Book for Writing Mathematical Proofs?

    Hello Could anyone recommend a good introductory book for learning how to write mathematical proofs. Thank you.
  30. K

    Are Calculus and Linear Algebra prerequisites for Intro to Mathematical Proof?

    Hi, all Going to take the Intro to mathematical proof class this coming quarter. It says the prerequisite is Cal III and linear algebra. I have taken those prereq years ago and forgot most of the materials. I don't know if the Intro to mathematical proof class has a lot of materials from...
  31. D

    Beat frequency- universal mathematical proof?

    It is easily derived from trigonometric identities that cos(2*pi*f1*t)+cos(2*pi*f2*t)=2cos(2*pi*(f2+f1)*t)*cos(2*pi*(abs(f2-f1)*t)) which proves that superposition of two cosine wave will generate a beat frequency, but what about about a universal proof that applies to any kind of...
  32. matqkks

    How to Help Students Understand Mathematical Proofs

    I am having difficulty with students understanding mathematical proofs. Even if they do understand them in the class they cannot reproduce them in the final examination. Majority of the students avoid doing the proof questions and concentrate on the numerical or technique application of...
  33. matqkks

    Motivate Students to Understand Logical Deduction of Proofs

    I have found that for a number of students doing proofs is an unnecessary evil. For some of them it is like a painful visit to the local dentist. I thought I had explained the need for proves but it still puts a number of students off. What motivating reasons can I give students for them to at...
  34. A

    100% Mathematical Proof vs Vellmen's How to Prove it

    "100% Mathematical Proof" vs Vellmen's "How to Prove it" Hello, I'm looking for books that teach proof methods and techniques. I know Vellemen's book is a popular choice but a few dissenting reviews among unanimous praise, on its amazon page, caught my attention. Like this one: And here are...
  35. M

    Is g(f) One-to-One if f is Not One-to-One? Prove It.

    [b]1. Assume that f : A -> B and g : B -> C and that f is not one to one. Prove that g(f) is not one to one. [b]2. one-to-one: x1 != x2 and f(x1) != f(x2) not one-to-one: f(x1)=f(x2) and x1 != x2 [b]3. The start of the proof should go as follows: Assume f: A->B, g: B -> C, f is not...
  36. F

    Linear Algebra + Mathematical Proof - Summer

    I was planning to take differential equations with Linear Algebra, but there is a conflict so I can't take both. Finally there is a course called Introduction to Linear Programming In my Physics/Math Calendar, it doesn't say I have to take it, so I am wondering what is it really...
  37. M

    Proof that (x^t)Ax>0 if Eigenvalues of Matrix A > 0

    I am looking for a rather simple proof that if the matrix A has eigenvalues>0, then (x^t)Ax>0 for any vector x not 0. My first tought was if the eigenvalues are bigger than 0, then (x^t)Ax=(x^t)"eigenvaulue"x="eigenvalue"(x^t)x>0, if x is nonzero and eigenvalues is bigger than 0. Is this...
  38. M

    Lock problem -need a mathematical proof

    Hi, There is a 2x2 matrix lock with one key in each cell. A key is nothing but a straight lever which can either be either in vertical or horizontal state. Please see the attached picture. The lock opens only when all the levers are horizontal. However, when you change the state of the lever...
  39. H

    Mathematical Proof of String Theory

    I have only taken a few college physics courses, and I have watched some videos about string theory. I would like to get excited about M-theory, but I have always wondered, why there wouldn't be an infinite number of ways to unite the forces, in which case M-theory is just one possibility...
  40. G

    Piston diameter size, fricition, mathematical proof

    I have already asked the question of piston diameter size and efficiency in an earlier thread but i'll recap. My question was that is it true that the larger diameter piston have a larger propensity to overcome the mechanical loss of fricition due to the piston rings contact with the mating...
  41. T

    Beginner's mathematical proof / composition of relations

    Homework Statement Suppose r and s are two positive real numbers. Let Dr and Ds be defined as in part 3 of Example 4.3.1. What is D_r \circ D_s? Justify your answer with a proof. (Hint: In your proof, you may find it helpful to use the triangle inequality.) Homework Equations Example 4.3.1...
  42. T

    What are the possible values of a in the inequality a < 1/a < b < 1/b?

    Homework Statement I originally made this thread for something else, but I have another problem that I need help with. Suppose a and b are nonzero real numbers. Prove that if a < 1/a < b < 1/b then a < -1. Homework Equations A hint was given for the problem: Assume a < 1/a < b <...
  43. F

    Is Magic Merely Unknown Science in Disguise?

    Let's say all things are known in the Universe and magic doesn't exist, then: K is the set of all things known and Magic, M, doesn't exist. M \notin {K} Let {\neg K} be all things not known. {\neg K} \notin {K} Since Magic can not be defined by {K} Then M \in {\neg K} by...
  44. P

    Validity of Mathematical Proof of Uncertainty Principle

    I saw a rather easy proof of the Heisenberg Uncertainty Principle in a PDE textbook the other day, but I'm not sure if it's correct. The proof goes as following: Note that \left| \int xf(x) f'(x) \right| \le \left[ \int |xf(x)|^2 dx \right]^{1/2} \left[ \int |f'(x)|^2 dx \right]^{1/2} , by the...
  45. M

    Iota Squared: Mathematical Proof and Physical Meaning

    i want the mathematical proof of iota squre = -1 , also what does it mean physically .
  46. U

    Beyond the Standard Model: Math Proofs for Year 11s

    Hi there, I am giving a talk to my class entitled "Beyond the standard model" - guess where I got that name from :p And I was wondering if there are any interesting mathematical proofs I can give to a group of Year 11's (GCSE) who don't have a huge amount of mathematical knowledge. I'm going...
  47. Y

    Mathematica Mathematical proof of the Big Bang Theory

    Can anyone tell where I can find the mathematical proof of the Big Bang theory and by whom, I'm interest in the calculation only. If you have a scientific paper available with you please send it to me(yousifhot@hotmail.com) I really need it. Thanks a lot,
  48. R

    Mathematica Math Proof Homework: Proving a_1 + a_2 + ... + a_K > K

    Homework Statement The thing is that there is a question which needs me to prove something. I have done it already but the thing that troubles me is that it wants me to prove this: a_{1}+a_{2}+a_{3}+....a_{K}>K a_{i} is a positive integer. I know this is always true. but how should I...