What is Proofs: Definition and 698 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.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Statistics: Proofs and Problems for Random Variables and their Distributions

    Homework Statement Before I get started here I have one really quick basic question: Lets say I want the probability that an survives two hours, and that the probability an engine will fail in any given hour is .02. Then I can get 1 - .02 - .98(.02) = .9604. This is found by a geometric...
  2. W

    Cholesky Factorization Proofs

    Homework Statement Let A =[A11 A12; A*12 A22] be Hermitian Positive-definite. Use Cholesky factorizations A11 = L1L*1 A22 = L2L*2 A22-A*12 A-111 A12 = L3L*3 to show the following: ||A22-A*12 A-111 A12||2≤||A||2 Homework Equations The Attempt at a Solution Using the submultiplicative and...
  3. N

    Should I Study Geometrical Proofs for Derivatives of Cos and Sin?

    For proofs such as the derivative of cos or sin.. should I study them both analytically and geometrically? By analytically I mean to derive them by algebraic means. Or should I also study the geometrical "intuition" behind it? I love proofs but aren't completely fond of the geometrical...
  4. M

    Question concerning rigor of proofs

    I've just started Spivak's Calculus and I'm having a few questions concerning the validity of certain of my proofs since some of mine are not the same as the ones in the answer book. Homework Statement Here is one of the proof: I need to prove that (ab)^{-1} = (a)^{-1}(b)^{-1}...
  5. A

    Separate proofs for discrete and cont. rv. cases of E(X-mu)^4

    Homework Statement X is a random variable with moments, E[X], E[X^2], E[X^3], and so forth. Prove the following is true for i) X is discrete, ii) X is continuous Homework Equations E[X-mu]^4 = E(X^4) - 4[E(X)][E(X^3)] + 6[E(X)]^2[E(X^2)] - 3[E(X)]^4 where mu=E(X) The Attempt at a...
  6. S

    Proofs by induction on immediate predecessors (well-formed formula complexity)

    Hi physics forum, I have no idea where to start with this: As far as I know the general pattern for this sort of proof is, 1) All atomic well-formed formulas (wffs) have some property P 2) From the assumption that immediate predecessors of any non-atomic wff A have P, so too does A...
  7. S

    Proofs by induction on immediate predecessors (well-formed formula complexity)

    Hi physics forum, I have no idea where to start with this: As far as I know the general pattern for this sort of proof is, 1) All atomic well-formed formulas (wffs) have some property P 2) From the assumption that immediate predecessors of any non-atomic wff A have P, so too does A...
  8. J

    Field Proofs (just needs revision)

    Homework Statement Thanks to everyone who has helped me so far - I'm very grateful. (1) Prove that the multiplicative inverse in any field is unique (2) Prove the cancellation law | ab = ac => b=c (3) Prove (-1)a = -a Homework Equations The field axioms...
  9. F

    Understanding vector calculus proofs

    ive been trying to understand a few of the identities my professor gave me and i can get a few of them down such as \nabla(\vec{A}\vec{B})=\vec{B}\nabla\vec{A} - \vec{A}\nabla\vec{B} and i can break it down through cartesian and product rules but when i try to do \nabla X (\vec{A}ψ) =...
  10. LiHJ

    Geometrical Proofs Homework: Get Expert Help Now

    Homework Statement Dear Mentors, Please guide me in solving the circled questions on the 2 attachments. Homework Equations Thank you The Attempt at a Solution
  11. LiHJ

    Geometrical Proofs: Get Expert Guidance Now

    Dear Mentors, Please guide me in solving the circled questions. Thank you
  12. N

    How to Ease my Way Into Proofs?

    I was very discouraged when I couldn't do a couple proofs myself in calculus such as the squeeze theorem. My textbook has very little steps into some of the proofs and assumes that the student should infer most of the information. Not being able to follow the proofs made me feel that I hated...
  13. K

    Solving Math Proofs: Get Help Quickly!

    1. prove that if 0<a<b, then a<\sqrt{}ab<a+b/2<b 2. \sqrt{}ab\leq(a+b)/2 holds for all a,b \geq 0 [b]3. Where do I begin? I have no clue! Thank you to anyone who can help!
  14. O

    Proof by Induction - Divisibility Proofs

    Homework Statement Q. Prove by induction that... (please see attachment). Homework Equations The Attempt at a Solution The end result should be divisible by 6, but hasn't worked out for me. Can someone help me spot where I've gone wrong? Thank you.
  15. A

    Matrix Homework: Solving for B in Statement 5

    Homework Statement See question 5 Homework Equations The Attempt at a Solution For part a, it is very easy. Multiply the inverse of A 2 times on both side, we can see the B=inverse of A. i.e. The required B is inverse of A, then the proof is finished. But how about part b...
  16. D

    Hausdorff Distance Proofs

    Homework Statement Given a compact set A\subset\Re^{n} and a point x\in\Re^{n} define the distance from x to A as the quantity: d(x, A)=inf({\left\|x-y\right\|: y\inA}) Given two compact sets A, B \subset\Re^{n}, define the Hausdorff distance between them to be: d(A, B)=max(sup{d(x, B) ...
  17. J

    Good book for calculus proofs?

    Hey, I know that questions about learning materials (like books) are supposed to be posted in the learning materials category but for some reason it is saying i cannot post there (it is saying i do not have privilieges/am trying to access administrative stuff?)...so i will just ask here...
  18. S

    Set builder notation and proofs

    "set builder" notation and proofs I'm curious about the references to "set builder" notation that I see in forum posts. Is this now a popular method of teaching elementary set theory and writing elementary proofs? I haven't looked at materials for that subject in the past 20 years. The...
  19. E

    Modular Arithmetic proofs (multiplication and addition mod n)

    Homework Statement Let n be a fixed positive integer greater than 1. If a (mod n) = a' and b (mod n) = b', prove that (a+b) (mod n) = (a'+b') (mod n) and that (ab) (mod n) = (a'b') (mod n) Homework Equations When a = qn + r a mod n = r The Attempt at a Solution (a'+b') (mod n) = (a...
  20. Z

    GCD Proof Check: 2 Problems Involving GCDs in Z

    I was doing a couple proofs (since I'm new to them) involving gcds and I just would like you guys to check them to see if I actually proved anything. There are 2 separate problems here. For both problems, a,b,c are in Z with a and b not both zero. PROBLEM 1 Homework Statement Prove...
  21. W

    Proofs with epsilon delta (real analysis)

    Hello, I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the what's and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however. Number 1. "Let f be a continuous function on the real numbers. Then the set {x in R ...
  22. Saladsamurai

    Spivak: Is his how to approach these proofs?

    Hello all :smile: I have started the problem set for Chapter one (basic properties of numbers) in Spivak's Calculus (self study). I think I am doing these right, but I have some questions. As a solid example, problem 1-(iv) says to prove the following: x^3 - y^3 =...
  23. S

    The FAQ on proofs should emphasize definitions

    I think the FAQ on proofs would be improved if it emphasized the use of defintions. It says that theorems and axioms are used in proofs, but many many textbook type proofs hinge on "parsing" definitions correctly. As alluded to in the FAQs related to "is .999.. = 1?", many difficulties that...
  24. O

    Solving ϵ-N Proofs: Simplifying the Denominator with sqrt(2)

    ϵ-N proof Homework Statement Homework Equations The Attempt at a Solution I've tried to make the denominator smaller as is usual with ϵ-N proofs. But the sqrt(2) confuses me. Any help is much appreciated.
  25. T

    Is my proof for (log10 a)/(log10 b) being irrational correct?

    I can never seem to create proofs the way it is shown in every textbook I've seen. To be honest, I don't really know how to write the proofs correctly. I've seen sometimes my reasons are flawed and other times I go around aimlessly and get home after some unnecessary steps. So I would just like...
  26. S

    What is the Best Book for Learning Proofs?

    Hey all, I have a pretty solid background in what would be best described as applied or engineering math. However, I have had a very limited exposure to proofs. This fall, I will be taking a course covering linear algebra and vector calculus in an entirely proof-based manner. I'm looking...
  27. F

    Preparing and Submitting Your Proof to Journals: Tips and Guidelines

    I believe I have proven a famous open problem in mathematics, and no, it is not the Riemann Zeta hypothesis although that would be nice. Anyway, I want to know how I can submit my proof online and if anyone can give me pointers on preparing my paper. Thanks.
  28. tiny-tim

    Geometry and trig proofs, with diagrams

    http://www.mathsisfun.com/geometry/" http://www.mathsisfun.com/algebra/trigonometry-index.html"
  29. I

    Becoming Fluent in Math Proofs - Tips & Advice

    I'm starting to learn how to write proofs, and I am wondering how to become fluent in proofs. Is it necessary to do problems that are IMO/Putnam? Can anyone give me some advice? Thanks in advance.
  30. D

    Is there a better way to present proofs by contradiction?

    Sometimes I find that while a proof can be carried out "by contradiction", this is a pretty sloppy way of proving the desired statement. I wonder if the "←" direction of the following proof is sound presentation of proof by contradiction. Statement. An integer is even if and only if its square...
  31. T

    Using Power-of-a-Point Theorem in Geometric Proofs

    Homework Statement Point A is on a circle whose center is O, AB is a tangent to the circle, AB = 6, D is inside of the circle, OD = 2, DB intersects the circle at C, and BC = DC = 3. Find the radius of the circle. Homework Equations Power of a point theorem (several cases found online...
  32. P

    How do I Write Proofs in French?

    I'm writing some of my proofs up in french in order to practice the language and I have a few questions for any of you french speakers out there. First, what voice is generally used 'nous' or 'on'. Second when I am telling the reader to perform this or that mathematical procedure, do I use...
  33. L

    What does f(x)>g(x) mean for x in [a,b]?

    Homework Statement Prove or falsify the statement (see picture) The Attempt at a Solution I've got the answer already but I want to make sure I know is what is meant by f(x)>g(x) for x in [a,b]. Does it mean f(x) lies above g(x) throughout the entire interval?
  34. I

    Calc I-III String: Dedicating Time to Proofs

    I'm currently in the middle of the Calc I-III string and I was wondering how much time I should dedicate to studying the proofs, if any time at all. I'm a physics major but I do plan on going on in math after I've taken all of the general curricula because I intend to pursue theory.
  35. A

    Solved] Proving Linear Transformation Properties with Linear Independent Sets

    Homework Statement Let V and W be vector spaces and T: V-> W be linear. a) Prove that T is one to one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W. b) Suppose that T is one to one and that S is a subset of V. Prove that S is linearly...
  36. 1

    Proofs for limits, feels unfamilar

    Firstly, I find the math syntax on this board incredibly difficult to use, so bear with me. Using any symbol makes the text appear on the next line... I don't know if it is my browser, or what, but I tried to make due. Sorry. Homework Statement Construct an "epsilon minus delta" proof for...
  37. M

    Identity Proofs of Inverse Trig Functions

    Homework Statement Prove the Identity (show how the derivatives are the same): arcsin ((x - 1)/(x + 1)) = 2arctan (sqr(x) - pi/2) Homework Equations d/dx (arcsin x) = 1/ sqr(1 - x2) d/dx (arctan x) = 1/ (1 + x2) All my attempts have been messy and it may be because I didn't...
  38. dkotschessaa

    Introduction to Proofs: A Beginner's Guide to Mathematical Logic

    I would like to start getting familiar with doing proofs, and I was wondering if someone could give me a good start. I am starting my "collection" in a sort of math notebook. Right now this is extra-curricular from my studies so I don't have time for anything complex. I would just like to...
  39. M

    Stirling numbers - hard proofs

    I have problem with prooving those two identities. Any help would be much appriciated! Show that: a) \begin{Bmatrix} m+n+1\\ m \end{Bmatrix} = \sum_{k=0}^{m} k \begin{Bmatrix} n+k\\k \end{Bmatrix} b) \sum_{k=0}^{n} \begin{pmatrix} n\\k \end{pmatrix}...
  40. P

    How can I handle long proofs in mathematics?

    I'm reading Calculus on Manifolds by Munkres and I often encounter multiple page proofs that are very technical. I can verify the argument in a reasonable amount of time, but to actually digest the proof (i.e. learn it such that I can reproduce it by memory weeks later) takes an inordinate...
  41. M

    Solve Trigonometry Proofs: Tan(x) – ½sin(2x) = tan(x)sin2(x)

    Homework Statement I need help with trigonometry proofs. the question asks me to prove the following and show all my steps. Prove that: Tan(x) – ½sin(2x) = tan(x)sin2(x)Homework Equations I don't know :( The Attempt at a Solution No attempt as I don't get it.Any help at all would be...
  42. J

    How can I improve my proof-writing skills?

    Don't get how to write proofs! I'm a high school student who really wants to major in mathematics. I love reading proofs, but when the book [What is Mathematics by Courant] asks me to do proofs, I have absolutely no idea of where to start. Should I just give up my aspiration to major in math...
  43. TheFerruccio

    Two Proofs for Statements a) and b) | Real Numbers, Exponential Inequalities

    I made two attempts at proofs. I feel the second one is ok, but the first one feels lacking. I'm not sure if I could represent it in a better way. Homework Statement Prove the following statements Homework Equations a) If x is real, and x > 1, then x^n > 1 b) If x is real, and x...
  44. M

    Proof of Aut(G): ϕ(Z(G))= Z(G)

    Homework Statement For every ϕ in Aut(G), ϕ(Z(G))= Z(G). Homework Equations Z(G):={g in G| gh=hg for all h in G} The Attempt at a Solution I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also...
  45. D

    Courant's Fundamentals, help with proofs

    Homework Statement a) For any fixed integer q > 1, prove that the set of points x = p/q^8, p, s ranging over all positive integers, is dense on the number line b) Show that if p is required to range only over a finite interval, p<= M for some fixed M, the set of all x is not dense on any...
  46. T

    How to Solve Logarithmic Equations Using Change of Base Formula?

    1) logba + logcb + logac = 1/logab + 1/logbc + 1/logca 2) logrp = q and logqr = p, show logqp = pq 3) if u = log9x, find in terms of u, logx81 4) log5x = 16logx5, solve for x attempt I know the change of base formula logax = logbx/logba, but I'm not sure if/how to apply it in any...
  47. M

    Where can I find proofs for d. eq solutions?

    Hello. Where can I find proofs for the solution of d. equations? I can find the solutions but I cannot find the proofs in any textbook. Specifically, how can I prove the solution for: y''+ay=0 y''+ay'+by=0 Thank you.
  48. C

    What are the steps to solve (sec∂-tan∂)²=(1-sin∂)/(1+sin∂)?

    [b]1. First one is (sin2x+sinx)/(cos2x+cosx+1)=tanx Second one is (sec∂-tan∂)²=(1-sin∂)/(1+sin∂) [b]2. Sec=1/cos tan=sin/cos cos²x+sin²x=1 [b]3. 1. I think eventually the sinx/cosx need to cancel to make tanx and the 1 could be used to create a lot of options 2. I have tried to...
  49. J

    Awe-Inspiring Math: The Most Beautiful Theorem Proofs

    What's the most Beautiful proof of a mathematical theorem you've seen?
  50. C

    Uniformly Bounded Functions: Proving Sequence Convergence

    Prove that a sequence of uniformly convergent bounded functions is uniformly bounded. Attempt at proof: So first we observe the following: ||fn||\leqMn. Each function is bounded. Also, |fn-f|\leq\epsilon for all n \geq N. First off, we observe that for finitely many fn's, we have them...
Back
Top