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. S

    MHB Formal Proofs in Maths: Establishing Equivalence

    From the book "FORMAL PROOFS IN MATHS "(Amazon.com),page 101 ,exercise19 ,Iread: Establish the equivalence between: 0<1,.........0<A\Longrightarrow 0<\frac{1}{A},............AC<BC\wedge 0<C\Longrightarrow A<B
  2. G

    A Algebraic Proofs of Levi-Civita Symbol Identities

    Hello everyone, my question concerns the following: Though widely used, there does not seem to be any standard reference where the common symmetrization and anti-symmetrization identities are rigorously proven in the general setting of ##n##-dimensional pseudo-Euclidean spaces. At least I have...
  3. M

    MHB Should NYC High Schools Ban Geometric Proofs?

    Most high schools in NYC have banned direct and indirect geometric proofs. Too many complaints from parents and schools in fear of a lawsuit decided to disregard proofs in geometry. Good idea? Bad idea?
  4. S

    Are proofs from calc class needed for EE?

    Hi, I am a first year calculus two student. I found that I have serious trouble understanding the derivation of formulas. For example, I can solve most of the exam problems because they are just using formulas, say arc length = Integral(sqrt( 1+ (dy/dx)^2)dx, but my textbook actually has two...
  5. Y

    MHB What Simple Theorems Can Be Proved Using Hilbert's Axioms of Geometry?

    Hello all, I am looking for simple theorems that can be proved by using Hilbert's axioms of Geometry only. For example, such a theorem can be "two lines intersect in a single point". I am looking for more examples that can be proved (with a short proof) using these axioms. Can you think of such...
  6. F

    Asymptote of a curve in polar coordinates

    Homework Statement The curve ##C## has polar equation ## r\theta =1 ## for ## 0<\theta<2\pi## Use the fact that ## \lim_{\theta \rightarrow 0}\frac{sin \theta }{\theta }=1## to show the line ## y=1## is an asymptote to ## C##.The Attempt at a Solution **Attempt** $$\ r\theta =1$$ $$\...
  7. D

    Mathematical proofs, physics and time management

    How important is for a physics undergraduate to know the mathematical proofs for every theorem learnd on the math courses? Is it better to trust the math, learn the intuitive notions, and then learn the methods and operations in a more mechanical way, memorizing formulas and steps through...
  8. A

    Studying How can I improve my ability to work with proofs?

    I'm a CS student and I'm about to take discrete mathematics next two semesters. My proofs are very weak and I want to change this. (I'm told discrete math is a lot of proofs.) Are there any books/courses/resources to help me work my way up? I have a summer to prepare for.
  9. S

    I Wrong proofs of Riemann hypothesis

    Hello! I read some stuff about the Riemann hypothesis and the formulation seems pretty clear. I also read that many proof of it (well basically all of them) are wrong. I was just wondering in which way are they wrong? (I haven't find a page with the wrong proofs, together with explanations of...
  10. T

    Relation closures proof

    Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...
  11. T

    I Proving Injectivity and Surjectivity: A Fundamental Concept in Function Theory

    Stumped on a couple of questions, if anyone could help! In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R. (a) Show that a function f is surjective if and only if there exists a function g such that f ◦ g = I. (b) Show that a function f is injective if and only if...
  12. S

    A Are there experimental proofs for modern theories

    Quantum theory, although hard to understand with intuition has a lot of experimental proof. Do the more modern theories e.g. String theory, or black hole theories have any experimental proof, or are they theories that the mathematics have led to? Without proof, do they deserve so much credit...
  13. C

    I Proofs of Stokes Theorem without Differential Forms

    Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it. I honestly will never use the higher dimensional version but I still want to see a full proof...
  14. N

    Intro Math Books for Learning Proofs | Jonathan

    Hello all! I am looking at books related to proofs, I have been looking around and it appears that either, "How it Prove It" or "Book of Proof" seem to be the top recommendations. Has anybody had any experience with these text and/or could provide some insight. Currently I am working through...
  15. Hypercube

    Mathematical physics - writing proofs

    Hi there! So for about a month, I've been self-studying mathematical physics using Mathematical Physics by Hassani. It's a big change for me, after only Stewart calculus, Boyce DE and some linear algebra course, but it's been loads of fun! Now, writing proofs is something fairly new to me, and...
  16. A

    I What's Wrong with the Proof?

    I am sure these are wrong given what is 'proven,' I just can't put my finger on it. Any insight into what is wrong is most appreciated. Thank you in advance!:1) Let q_1, q_2, q_3, … be an enumeration of A = \mathbb{Q} \cap (0,1).2) Select a real number r \in (0,1)...
  17. Kernul

    Proofs of the four type of intervals

    Homework Statement Verify that if an interval is bounded it must be one of the four following types: ##(a, b)##, ##[a, b]##, ##(a, b]##, or ##[a, b)##. Homework EquationsThe Attempt at a Solution I don't quite get what I should actually prove here. Do I have to see if, for ##A \subseteq S##...
  18. Mr Davis 97

    I Proving Theorems Not in "If-Then" Form

    My textbook goes into depth about proof techniques and about how to go about proving theorems. However, the author only really focuses on theorems that are stated in the form "if p, then q." I know that a great many theorems have this logical structure, so it is good to know how to prove them...
  19. Nipuna Weerasekara

    Can a triangle be formed with these length constraints?

    Homework Statement There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio. Is this statement true or false...? Homework Equations $$Golden \space Ratio = \phi = 1.618... $$ The Attempt at a Solution Actually I have no clue at all...
  20. F

    I Proofs of various integral properties

    I've been trying to prove a couple of properties of integrals using the Riemann sum definition: $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned (such that ##a=x_{1}<x_{2}<\cdots <x_{i-1}<x_{i}<\cdots...
  21. J

    Find the Union of Intervals: A_n

    Homework Statement Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n## Homework Equations What does that last statement mean? Union for n greater than or equal to one times the interval? The Attempt at a Solution I can't understand the question.
  22. J

    B Proofs: Hypotenuse is the longest side of a right triangle

    I want to prove that the hypotenuse is the longest side of a right angled triangle. Could people check that the proof I'm giving is correct? Say the hypotenuse is of length ##c## and the other two sides are of length ##a## and ##b##. First of all, we obviously have: ##a^2 + b^2 > a^2 \quad##...
  23. K

    Are you a Mathematics enthusiast studying at a University?

    I'm Krim and I am a Mathematics enthusiast. I am quite interested to other Science fields, too.
  24. PhysicsBoyMan

    I How to know if a polynomial is odd or even?

    3(2k+1)3 I have written a program which calculates the value of that polynomial with different values of k. The result is always an odd number. I am having a difficult time writing a proof that states that this polynomial always returns an odd number. I know that (2k + 1) is the general form...
  25. S

    MHB Proving p<=>~p: Two Contradictory and Identical Proofs Explained

    CANNOT FIGURE THIS OUT: proof No1 1. p<=>~p.........assumption 2. (p<=>~p)<=>[(p=>~p)^(~p=>p)]......definition of (1) 3. (p<=>~p)=>[(p=>~p)^(~p=>p)].....2,and Biconditional Elimination 4. (p=>~p)^(~p=>p).........1,3 M.Ponens 5. (~p=>p)............4, Simplification 6...
  26. J

    I Can you make the induction step by contradiction?

    Assuming you've sufficiently proven your inductive basis, can you complete a proof by induction in the following manner: Make the inductive hypothesis, assume P(n) is true for some n. Assume P(n+1) is not true. If it follows from the assumption that P(n+1) is false that P(n) must also...
  27. S

    Insights Irrationality for Dummies - Comments

    swamp-thing submitted a new PF Insights post Irrationality for Dummies Continue reading the Original PF Insights Post.
  28. A

    MHB Solving Logic Proofs: A Step-by-Step Guide

    hi! I've been studying formal proofs for my logic final and i had a question on this one that i found in my textbook: 1.)~P\supset(horseshoe)U 2.)P\supsetF 3.)F\supsetU\thereforeU i just have had a little trouble in the past getting started with proofs. Can anyone give me a little push and...
  29. O

    I Understanding the Logic of Quantifiers: A Guide for Mathematical Proofs

    I'm new to proofs and I'm not sure from which assumptions one has to start with in a proof. I'm trying to prove the generalized associative law for groups and if I start with the axioms of a group as the assumptions then I already have the proof. From what basic assumptions should one start...
  30. T

    B Debunking Alleged proofs against modern science

    Hello fellow PF'ers! I have been conversating with a strongly religious character for some time now, and thought i'd present some of his arguments to you. I have myself tried to explain the basics of the problems, but maybe some of you can have a laugh and go into further details as to why he...
  31. C

    I Understanding Number Theory Proofs: Order of Elements in Finite Groups

    I just want to make sure I understand these number theory proofs. b^{\phi (n)}=1mod(n) \phi (n) is the order of the group, so b to some power will equal the identity. so that's why it is equal to one. b^p=bmod(p) b^p=b^{p-1}b b^{p-1} produces the identity since p-1 is the...
  32. SrVishi

    Other Proof Tips for Math Majors: Logic & Techniques for Real Analysis

    Every math major eventually learns logic and standard proof techniques. For example, to show that a rigorous statement P implies statement Q, we suppose the statement P is true and use that to show Q is true. This, along with the other general proof techniques are very broad. A math major would...
  33. O

    Algebra Sheldon Axler's Algebra and Trig/Precalculus for proofs?

    Can someone tell me if Axler's texts use proofs? I'm looking for a book that teaches proofs at the high school level. Something other than a geometry text.
  34. B

    Analysis Seeking a Rudin's PMA-level analysis book with abstract proofs

    Dear Physics Forum personnel, I recently got interested in the art of abstract proof, where the focus is writing the proof as general as possible rather than starting with a specific cases. Could anyone recommend an analysis book at the level of Rudin's PMA that treats the introductory...
  35. Y

    Proofs involving Negations and Conditionals

    Suppose that A\B is disjoint from C and x∈ A . Prove that if x ∈ C then x ∈ B . So I know that A\B∩C = ∅ which means A\B and C don't share any elements. But I don't necessarily understand how to prove this. I heard I could use a contrapositive to solve it, but how do I set it up. Which is P...
  36. moriheru

    Foundations Proof of x^2n beeing even and other fundamental proofs

    Is there a book containing fundamental proofs such as any number of the form x^2n beeing even and such. I know this is very vague, so I must apologize. Thanks for any help.
  37. T

    MHB Beginner Exercises on Prime numbers and Equality / Inequality proofs

    Does anyone know a good resource for exercises on these topics?
  38. I

    Hermitian adjoint operators (simple "proofs")

    Homework Statement I'm having some trouble with questions asking me to "show" or "prove" instead of computing an answer so I'm looking for some input if I'm actually doing what I'm supposed to or not (and for the last one I don't know where to get started really.) 1. Show that ##T^*## is...
  39. H

    Common proofs? (studying for an exam)

    Homework Statement I'm currently studying for an exam, but I don't know exactly what could be on it - the specification is very vague. I was just wondering if there were some useful proofs I should know? I've covered things like proving the diffraction grating experiment, centripetal...
  40. M

    Linear Algebra Proofs for Engineering Majors: A Fair Assessment?

    I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The homework assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the...
  41. T

    Identify What's Wrong with the Argument (Logic and Proofs)

    Homework Statement Taken from Discrete Mathematics and its Applications, Seventh Edition: "What is wrong with this argument? Let S(x, y) be 'x is shorter than y.' Given the premise \exists s S(s, Max), it follows that S(Max, Max). Then by existential generalization it follows that \exists x...
  42. Multiple_Authors

    Insights Proofs in Mathematics - Comments

    Multiple_Authors submitted a new PF Insights post Proofs in Mathematics Continue reading the Original PF Insights Post.
  43. D

    Consequences of space-/time-/light-like separations

    I'm trying to prove the following statements relating to space-like, time-like and light-like space-time intervals: 1. There exists a reference frame in which two space-time events are simultaneous if and only if the two events are space-like separated. 2. There exists a reference frame in...
  44. I

    When do you begin to prove? which maths lead to proofs?

    Hi, Could someone please tell me at which point in learning maths do you begin to write and solve proofs? I have taken high school maths so far except for discrete math. If there is a list of different maths that lead up to proof writing, please let me know of them and in which order I...
  45. Nathanael

    Zero experience with proofs, does this count?

    Homework Statement I'm trying to show that ##\Sigma_j(\vec a_j \cdot \vec B)=\vec B\cdot\Sigma_j(\vec a_j)## (I need this to be true to derive some angular momentum properties.) 2. The attempt at a solution Let's say that in some coordinate system we can express the vectors as ##\vec...
  46. Ryanodie

    Check my work on these proofs? (basic set theory)

    Homework Statement [/B] I am going through Apostol's Calculus volume 1 and am working through I 2.5 #3. I'm not very familiar with doing proofs so I just wanted to make sure that I got the right idea here. Here's the question: Let A = {1}, B = {1,2} Prove: 1. ## A \subset B ## 2. ## A...
  47. Keen94

    Proofs using the binomial theorem

    Homework Statement Prove that ∑nj=0(-1)j(nCj)=0Homework Equations Definition of binomial theorem. The Attempt at a Solution If n∈ℕ and 0≤ j < n then 0=∑nj=0(-1)j(nCj) We know that if a,b∈ℝ and n∈ℕ then (a+b)n=∑nj=0(nCj)(an-jbj) Let a=1 and b= -1 so that 0=(1+(-1))n=∑nj=0(nCj)(1n-j(-1)j)...
  48. Keen94

    Induction Proofs Homework: Prove ∑nj=0 n C r = 2n

    Homework Statement Prove that ∑nj=0 n C r = 2nHomework Equations Defn. of a combination. Defn. of mathematical induction.The Attempt at a Solution The formula is true for n=1 2=∑j=01 n C r = 1 C 0 +1 C 1 =1 + 1 = 2 Now assume that for some k∈ℕ and 0≤ j ≤ k we have 2k = ∑kj=0 (k C j) Then...
  49. P

    Intro Math Introduction to Proofs Textbook Recommendation

    I'm planning on taking an Abstract Math and Linear Algebra II (essentially a more theoretical continuation to the first linear algebra course which all physics majors take) in order to later take courses like group theory, number theory, analysis, etc. But I have close to no knowledge with how...
  50. P

    Difficult number theory problem proofs

    The following is a repost from 2008 from someone else as there was no solution offered or provided I thought id post one here Homework Statement neither my professor nor my TA could figure this out. so they are offering fat extra credit for the following problem Let n be a positive integer...
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