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.

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  1. Z

    Do these two statements imply an underlying induction proof?

    Here is one proof $$\forall u\in U\implies Tu\in U\subset V\implies T^2u\in U\implies \forall m\in\mathbb{N}, T^m\in U\tag{1}$$ Is the statement above actually a proof that ##\forall m\in\mathbb{N}, T^m\in U## or is it just shorthand for "this can be proved by induction"? In other words, for...
  2. M

    Proofs about the second-order linear differential equation?

    Proof: (i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##. This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
  3. XcgsdV

    Other Is it worth doing EVERY problem in a textbook?

    I am a freshman Physics major currently working through Apostol's Calculus Volume 1 in my free time, somewhat to further develop my calculus knowledge, but mainly for fun. Apostol's text is proof-based, and as such has a number of problems that are just proofs. As a hopeful future Biophysicist...
  4. MevsEinstein

    I Proving Goldbach's Conjecture for Math Proofs

    Lets say you were trying to prove a math statement when you realize that you can use a conjecture (say, Goldbach's conjecture) to finish the proof. If you don't have the time or the brains to prove it, how many cases of Goldbach's conjecture do you prove so that you can use it in your proof?
  5. B

    I Explanation and clarifications in epsilon-delta limit proofs

    Good afternoon. I have some questions about the details of epsilon-delta proofs. Below is a simple, non-linear limit proof example which will serve as an example of the questions I have. The questions are below the example and involve clarification and explanations of steps and details in the...
  6. M

    Can anyone please review/verify my proofs for gcd problem?

    Proof: (a) Suppose that gcd(a, b)=1. Let d=gcd(a+b, a-b). By definition of the greatest common divisor, we have that d##\mid##(a+b) and d##\mid##(a-b). This means d##\mid##[(a+b)+(a-b)] and...
  7. docnet

    B How do mathematicians come up with new proofs?

    I watched an interview of Yitang Zhang and he said "the way to prove a finite limit of bounded gaps between primes came to him during ##30## minutes in an afternoon", and he worked alone and did not collaborate with others during his research time. After looking up the proof, I am in disbelief...
  8. M

    I Question regarding writing proofs

    I have a couple general questions regarding writing proofs: Do proofs typically fall into being one out of all of the rules of inference (page 6-7 on this pdf) or is it that generally, most proofs may categorically qualify within a very small subset of the rules of inference (say “many common...
  9. Mikaelochi

    I Doing proofs with the variety function and the Zariski topology

    I included this image because it is easier than typing it out. Anyway, this is an old problem I need to catch up on. I have a clue as to how to do part a. I could say given an x that is a member of ∩V(Ai) which implies that x is a member of V(Ai) for ∀i. Then we can say ∀i all polynomials are in...
  10. N

    Computational Looking for a Geometry Proofs Textbook?

    I am seeking a geometry proofs textbook. In other words, I seek a textbook that shows all geometric proofs from start to finish. There are books that show proofs worked out as a reference book for students. Can someone provide me with a good geometry book for this purpose? I am particularly...
  11. C

    Changing the Statement Combinatorial proofs & Contraposition

    I have a question regarding to combinatorial proofs and predicate logic. It seems to me that in some combinatorial proofs there is a use of contraposition ( although not explicitly stated in the books where I've read so far ), for example If we to prove that ## C(n,k) = C(n,n-k) ##...
  12. Constructivist

    Analysis Understanding how they thought of math proofs in the first place

    Hi,I am learning pure math say real analysis from Rudin. Now, I am getting confused on philosophy of learning. Well, if I read Rudin, somehow with struggle, I am able to fill the gaps in the proof which he gives. Though I can completely understand the proof, my question is how was the proof...
  13. J

    B Are proofs needed for definitions? Conditional probabilities

    My probability class has me wondering about pure math questions now. We started with the axioms and are slowly building up the theory. Everything was fine but then a definition of Conditional Probability P[A|B] = \frac{P[AB]}{P} appeared and it's just not sitting right with me. I know that...
  14. C

    Simple Induction Direct Proofs regarding Induction

    Summary:: . When asked to prove by Induction, I'm asked to prove a statement of the form: Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating...
  15. fresh_42

    Your favorite Eureka moment for proofs

    What was your favorite proof or Eureka moment you remember? Mine was about normal subgroups (##gNg^{-1}\subseteq N##). You learn the definition and prove a few properties and go to the next subject. I remember that I once tried to teach the concept to someone and in the middle of my...
  16. R

    Intro Math Suggestions for an Intro to Proofs Math textbook for Self Study

    Hi everyone. I'll be taking a course in the fall called "Introduction to Mathematical Reasoning", which is basically and introduction to proof-based mathematics. The syllabus says we will be covering the first seven chapters of Smith, Eggen, and St. Andre's "Transition to Advanced...
  17. LCSphysicist

    Proofs in analytic geometry and vector spaces.

    I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about: For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?: d1 = Diagonal one = (a,b,c) d2 =...
  18. E

    MHB Can we prove that (m+1)/(n+1) > m/n if n>m>0 using synthetic proof?

    Let m,n be real numbers. Prove that if n>m>0 , then (m+1)/(n+1) > m/n I'm currently confuse in this one help will be very much needed
  19. SamRoss

    Do mathematicians still work on proofs by hand?

    With automated theorem proving, what is left for mathematicians other than perhaps inputting weird axioms? Or are the machines not as sophisticated yet as I'm assuming they are?
  20. DaTario

    I Compilations of proofs of Euclid's Theorem on primes

    Hi All. Does anybody have a reference, (book, internet site) - besides those books of Paulo Ribenboim - where one can find a compilation of demonstrations of the Euclid's theorem on the infinitude of primes? As a suggestion, if the known proofs are neither too many not too long, it would be nice...
  21. J

    MHB Find Proofs for the following 5 propositional logic statements

    i came acroos the below while studying propositional Logic, can anyone find the proofs 1) P ⊢ P 2) P → Q, Q→R ⊢ P → R 3) P → Q, Q→R, ¬R ⊢ ¬P 4) Q→R ⊢ (PvQ) → (PvR) 5) P →Q ⊢ (P&R) → (Q&R)
  22. hackedagainanda

    Foundations Looking for Introduction to Proofs

    Hey, PF! I am currently studying Algebra and Trig, and was wondering what's a good book to use to get eased into the process of proving statements and theorems. I'm planning to use Basic Mathematics by Lang but the proofs in it have been a real hindrance to my progress. Even just simple proofs...
  23. J

    Algebra Book on how to write proper proofs in Group Theory

    I am trying to learn group theory on my own from Schaum's Outline of Group Theory. I chose this book because there are a lot of exercises with solutions, but I have several problems with it. 1) In many cases the author just makes some handwavey statement and I have to spend hours or days trying...
  24. Cerenkov

    B Examples of theoretical proofs overturned by evidence?

    Hello. https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1970.0021 The Singularities of Gravitational Collapse and Cosmology The above paper by Hawking and Penrose is presented in the form of a mathematical proof. To my knowledge the predictions it makes about the initial cosmological...
  25. nomadreid

    I Are existence proofs outlawed in multi-valued logics?

    First, is my assumption that all consistent multi-valued logics obey the principle of explosion from a false proposition correct? If so, how would one prove that? (I assume it is, because if not, then by the definition of intuitionist logic by Wolfram Mathworld...
  26. N

    I Exploring the Benefits of Epsilon-Delta Proofs in Mathematical Analysis

    I understand the concept of Epsilon-Delta proofs, but I can't understand why we have to do them. What's the advantage of using this proof over just showing that the limit from the function approaches from the left and right are the same?
  27. A

    MHB Robinson Arithmetic (Q) Proofs

    In Peter Smith's Godel book, 2 conditions are proven of several that make Q "order adequate" O2: For any n, Q ⊢ ∀x ({x=0 v x=1 v...v x=n} → x≤n) 03: For any n, Q ⊢ ∀x (x≤ n → {x=0 v x=1 v...v x=n}) O3 is proved by induction. O2 is not. It would appear as if induction would be required in...
  28. matqkks

    How to motivate students to do proofs?

    I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this. How can I inspire them to love essential kind of mathematics? They love doing mathematical techniques...
  29. wrobel

    Insights A Pure Hamiltonian Proof of the Maupertuis Principle - Comments

    Greg Bernhardt submitted a new blog post A Pure Hamiltonian Proof of the Maupertuis Principle Continue reading the Original Blog Post.
  30. V

    Proofs By Induction: Proving P(n) for All n

    Homework Statement Suppose we want to prove using mathematical induction that for all positive integers n, 12+22+...+n2 = (n(n+1)(2n+1))/6. What do we need to prove in the inductive step of our proof? Homework EquationsThe Attempt at a Solution I am struggling to understand what this means...
  31. V

    Proofs Homework: Explain Why m+n≠10 Example Not Sufficient

    Homework Statement Suppose you are asked to prove that for all integers m and n, m+n≠10. You give the example m=1 and n=2. Why is this not sufficient? Homework EquationsThe Attempt at a Solution I can't quite understand why it is not sufficient? Could someone please explain to me why is it...
  32. S

    I Formal proof for the theorem of corresponding angles

    Recently I started looking back at some basic mathematical principles, and I started thinking about the theorem of corresponding angles. It's such a basic idea that it seems obvious on an intuitive level, but despite that (or possibly because of that) I can't think of a good way to formally...
  33. T

    Showing ##\sqrt{2}\in\Bbb{R}## using Dedekind cuts

    1. The problem statement, all variables and given Prove that ##\sqrt{2}\in\Bbb{R}## by showing ##x\cdot x=2## where ##x=A\vert B## is the cut in ##\Bbb{Q}## such that ##A=\{r\in\Bbb{Q}\quad \vert \quad r\leq 0 \quad\lor\quad r^2\lt 2\}##. I believe that I have to show ##A^2=L## however, it...
  34. L

    Programs Chemistry-Based Research Program or more Math/CS Courses?

    To sum up the current biggest stress of my life into a question: do you recommend that I stay in a biology/chemistry-based research program or pursue proof-based math courses? I am a freshman physics major at a good research university. I was invited into a very difficult program for "future...
  35. karush

    MHB What is the Equivalence Class for a Fixed Integer in Hurricane Lane's Aftermath?

    ok need help with these 3 questions I know its fairly easy but its new to me, so we have had hurricane Lane here this week but mahalo much
  36. T

    I Writing proofs that are more or less formal

    How do I know if a proof I am writing or reading is informal or formal enough? Of course there are obvious distinctions like a formal proofs cannot constitute a drawing (e.g. venn diagrams, triangles for the triangle inequality), but sometimes I read proofs that uses phrases like "Continue in...
  37. opus

    B Doing proofs: Setting an expression as a variable

    In a book I'm reading, I'm told to prove that: If m and n are odd, then (m)(n) is odd. The proof goes as such: Let m=(2a+1) and n=(2b+1) Then, mn= (2a+1)(2b+1) = 4ab+2a+2b+1 = 2(2ab+a+b)+1 = 2t+1 ; where t= 2ab+a+b Two questions: When we take an expression, and assign it to a single...
  38. G

    Analysis Is this a good textbook to learn proofs?

    Hey guys, I was wondering if this textbook is good. I've seen the reviews and it has high stars. I want to learn proofs so I self-study and make my way to finally learning real analysis. I'm a chemistry major with a concentration in Math and economics. My plan was to do a double major in math...
  39. Rin

    Algebra Books Best for Mathematics & Algebra Self-Study with Proofs?

    Hello, I've been trying to improve my algebra since I've never been particularly good with math. I'm going through Serge Lang's Basic Mathematics textbook and while I have been learning a lot his proof-based exercises are a pain to get through and the back of the book only provides answers for...
  40. F

    How Does Choosing N Affect Limit Proofs in Calculus?

    Homework Statement Find and prove ##\operatorname{lim} \frac {1}{n^2}##. Homework Equations In the textbook, we assume that the limit is going to infinity without writing it. If L is the limit, we have for all ##\epsilon > 0##, there exists ##N## such that ##n \epsilon \mathbb{Z}## and ##n >...
  41. D

    B Can multivariate non-negative polynomials always be written as a sum of squares?

    Does a proof by counterexample belong to direct or indirect type of proof?
  42. D

    LaTeX Sample problems; Simple Limit (Epsilon - Delta proofs) Latex code included

    Remember to use the appropriate packages; these are in similar post if a mod wants to add the link if you choose to use Latex. Here is the PDF \begin{document} \begin{center} {\LARGE Epsilon-Delta Proofs \\[0.25em] Practice} \\[1em] {\large Just for practice, don't use Google to cheat!}...
  43. D

    MHB Sample problems for MVT/Integral proofs (elementary level calc)

    Note the code requires the appropriate package in latex given here; just don't forget to include you document class in the preamble of the document. \usepackage{amsmath, amssymb, amsthm} \usepackage{graphicx} \newtheorem*{thm}{Theorem} \renewcommand{\qedsymbol}{${\scriptstyle...
  44. Greg Bernhardt

    Proofs and facts in science, math and life

    Warning! Laymen reasoning ahead :) I've long heard that in science nothing is proven. That proof is a mathematical term. So what exactly do you call a certifiably known fact? An example is that the Earth is round or at least that the Earth is not flat. What do you call that? Can we not say it's...
  45. mr.tea

    I Help Needed: Understanding Hungerford's Algebra Book Proofs

    I am trying to learn about free groups(as part of my Bachelor's thesis), and was assigned with Hungerford's Algebra book. Unfortunately, the book uses some aspects from category theory(which I have not learned). If someone has an access to the book and can help me, I would be grateful. First...
  46. CollinsArg

    Fundamental Proofs in General Physics?

    Hello, I'm a freshman and I'm struggling with some questions. I'm trying to relate all I'm learning from my Physics classes to the real world (that is the mathematical connection and its expanations with the real world). I have some especifics questions but also a general answer I think could be...
  47. S

    Upper & Lower Bounds Real Zeros-Polynomial Division, Proofs?

    Homework Statement Upper Bound[/B] If all of the numbers in the final line of the synthetic division tableau are non-positive, prove for ##f(b)<0##, no real number ##b > c## can be a zero of ##f## Lower Bound To prove the lower bound part of the theorem, note that a lower bound for the...
  48. Mr Davis 97

    Does a Smallest Real Number Exist for a Given Real Number?

    Homework Statement (1) Prove that there exists no smallest positive real number. (2) Does there exist a smallest positive rational number? (3) Given a real number x, does there exist a smallest real number y > x? Homework EquationsThe Attempt at a Solution (1) Suppose that ##a## is the...