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. M

    How can I prove that f + g is convex?

    hey everyone: Use a definition to work forward from each of the following statements. b. for functions f anf g the function f + g is convex, where f + g is the function whoes value at any point x is f(x) + g(x). Definition of a convex function...
  2. T

    Levi-Civita proofs for divergence of curls, etc

    I've also posted this in the Math forum as it is math as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B}) b) \nabla...
  3. T

    Levi-Civita proofs for divergence of curls, etc

    I've also posted this in the Physics forum as it applies to some physical aspects as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times...
  4. N

    How to Approach Proofs in Mathematics: Tips and Strategies

    Homework Statement If x and y are arbitrary real numbers with x<y, prove that there is at least one real z satisfying x<z<y Homework Equations The Attempt at a Solution The problem arises from my inexperience in rigorously proving anything. If possible a general explanation of...
  5. A

    Proofs Involving Greatest Common Divisors

    I'm not sure if it goes here or the section beyond calculus, so I'm just putting it here because it doesn't involve any calculus. Homework Statement Suppose that (a,b)=1 [Greatest Common Divisor=1] and (a,c)=1. Does (bc, a)=1? Homework Equations (a,b)=d=au+bv, where u and v are...
  6. I

    Help with cross product proofs

    I have a 23 problem assignment due at the end of the week, and although I'm going to have a chance to talk to my teacher about the questions I have, I'd like to go ahead and get going on the problems. I've successfully completed 21 of them, but the last two are stumping me. I'm submitting them...
  7. S

    Proofs involving functions and sets. Related questions.

    Hey everybody... I have a few quick questions concerning sets and functions for the experts... I've been trained in applied mathematics, so I'm not really used to this kind of formalism. 1. Can somebody look at my "proposed proof" of this elementary theorem for me? I have a feeling that it...
  8. S

    Philosophy of basic set theory proofs involving or .

    Philosophy of basic set theory proofs involving "or". Hey! I'm working through an Introduction to Analysis text, and I'm currently on the first chapter, which covers set theory. In one of the end-of-chapter problems, I'm asked to prove a basic theorem which leads to the following statement...
  9. S

    Learning Math Proofs in College: A Guide for New Students

    Hi, I finished calculus 1 in college this past year, and I was reviewing it in the summer to make sure I understand it and have a solid foundation for when continue taking math classes this upcoming year. My math has always been lacking a little from my high school past where I never paid...
  10. F

    How Can I Improve My Proof Writing Without Feedback?

    My adviser asked me to study the first 50 pages of a book so I'm working the exercises. But they are all proofs, so I have no idea if I'm doing them correctly. I can't find any answers online, and even if I did, that would just tell me one way of proving it-- and there are, of course, many...
  11. I

    Is regularity preserved in subsets of regular spaces?

    i've texed up three proofs in from elementary topology. can someone please check them? actually i'll just retype them here for convenience 8.2.5 Let f: X_{\tau} \rightarrow Y_{\nu} be continuous and injective. Also let Y_{\nu} be Hausdorff. Prove : X_{\tau} is Hausdorff...
  12. 3

    Proof of U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y) using sqrt property

    Homework Statement For U(w)=sqrt(w), prove that U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y) Homework Equations sqrt(x)=x^(1/2) The Attempt at a Solution I have: sqrt(pie(x)+(1-pie)y) > pie*sqrt(x)+(1-pie)sqrt(y) so... (pie(x)+(1-pie)y)^(1/2) > pie*(x)^1/2+(1-pie)(y)^1/2...
  13. Repetit

    Constructing Proofs in Mathematics: Where do I Begin?

    Hey Im trying to study abstract algebra, set theory and group theory, on my own. I have trouble understanding how to construct mathematical proofs though. Some of the things the excercises tells me to prove, seems so intuitively clear and obvious that I don't know what's left to prove. For...
  14. M

    Proving Set Theory Statements: A Guide for Beginners

    I've been working on these problems and unfortunately i can't make heads or tails of these two. Any insight where to start the proof would be great. 1)Let A, B and C be sets. Show that if A~B⊆C, then A~C⊆B holds. What I got so far: Is it correct to state that A~B = A⋂B' and A~C = A⋂C'...
  15. T

    Learning Logic to Master Math Proofs

    Would taking an intro to logic course help me prepare for the abstract proof writing skills that I'll need in upper division math?
  16. D

    Epsilon-Delta Definition of Limit (Proofs)

    In my self-study Calculus book I finished with the 'intuitive' definition of the limit and the text directed me to the 'formal' definition of the limit. After reading the section covering it a few times I think I comprehended the details of the rigorous rules dictating it - but obviously not...
  17. W

    Trouble understanding the proofs in Marion and Thorton

    Homework Statement I am having trouble understanding the proofs in Marion and Thorton [Newest Edition]. The section where he goes through proof of products in tensor notation. An example is page 26 example 1.6. I don't get the switching of the indices on the very last part. Also can someone...
  18. D

    Epsilon-Delta proofs, once again

    I'm trying to understand \epsilon-\delta proofs, but I'm having some trouble. For example, if we want to prove that \lim_{x\rightarrow2}x^3=8, starting from |x^3-8| we get to something like |x-2||x^2+2x+4| And this is what confuses me: we conjecture that |x-2|<1, then |x|<3, so we get...
  19. H

    What are the Funniest Proofs in Geometry?

    http://www.themathlab.com/geometry/funnyproofs.htm
  20. O

    Couple of Proofs (Regular Induction / Well Ordering)

    [SOLVED] Couple of Proofs (Regular Induction / Well Ordering) Hi there everyone, I've been having a bit of trouble of solving these questions, so any help would be greatly appreciated: Homework Statement 1: Prove, via regular induction, that it is possible to draw a line-segment of length...
  21. G

    How do I improve my skills in constructing mathematical proofs?

    Hi First of all, I would like to mention that I can do proofs that involve algebraic manipulations (in a field i.e.) pretty well, or proofs that involve epsilon-delta arguments or mathematical induction. However, at the moment I'm reading "Principles of mathematical analysis" and I have a hard...
  22. B

    Comparing Books on Writing Proofs: Which to Choose?

    I searched around and I found some books on how to write proofs. There are so many of them that got good review and I have no idea which to choose. Here are some books I am considering: How to Solve It, by Polya An Introduction For Mathematical Reasoning, by Eccles The Nuts and Bolts of...
  23. S

    Proofs involving Catalan Numbers

    Homework Statement I need to prove two things about the Catalan numbers. The first is that Cn is odd iff n=(2^k)-1 for some positive integer k. The second is that given the matrix A defined by the rule a(i,j)=C(i+j), prove that det A=1. I have not covered determinants in my linear class...
  24. B

    Proving AD is congruent to AE: How to Use Triangle Congruence Proofs

    Homework Statement ......A..... ....../\..... ....../..\... ...../...\.... ..../...\... .../...\... .../...\... ...../...\..... ..../...\.../.\.... .../...\.../...\... .../...\..F./...\... .../...5..x..6...\... .../.../...\...\... .../.../...\...\.. .../.../....\...\...
  25. E

    Proof of Unit Circle: AE = Tan(\theta)

    Homework Statement The problem comes with a diagram but I'll use the wikipedia diagram because it's nice and pretty and I'll just rearrange the letters to suit it. http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg Just in case the image doesn't load in the page...
  26. A

    Proving Limsup h.w. Proofs: Monotonic Increasing Function

    Homework Statement 1. A function f(x) is said to be monotonic increasing in A if for all x1, x2 ∈ A, x1≤x2 implies f(x1)≤f(x2). Prove that if f(x) is monotonic increasing in R [f: R→R] and c is a cluster point of R then the limit of f(x) as x→c^{-} exists (might be +∞). 2. s(δ) =...
  27. S

    Some Composition Proofs for Surjectivity and Injectivity

    [SOLVED] Some Compostion Proofs Homework Statement Prove: 1.) The composition of subjective functions is subjective 2.) The composition of injective functions is injective Homework Equations Subjective: A function f: A->B is surjective iff For all members of B, there exists a...
  28. O

    Help with 2 Proofs: Prove & Show Uniqueness

    Hi there everyone, I have the basic idea of what to do, its just trying to show the cases work is where the problems occurs. Anyways for the first one: Homework Statement Prove that if x is any positive integer, then ⌈x/2⌉ ≤ (x + 1)/2. (Here, for any real number r, ⌈r⌉ is the smallest...
  29. E

    Challenges with Induction Proofs: Strategies and Solutions

    hello, i am having some trouble with a few induction problems. 1) Prove by induction that for all natural numbers n, n^2 + 3 < 2^n + 5. 2) Prove by induction that for all natural numbers n, (1-1/2)(1-1/4)...(1-1/(2^n)) > or equal to 1/4 + 1/(2^(n+1)). i got started on these but ran into...
  30. S

    Proving Induction and Divisibility - Two Simple Homework Problems

    Homework Statement Prove 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all natural numbers n. Homework Equations The Attempt at a Solution Well, this seems like the typical induction proof, so I start by testing the hypothesis at 1: 1^3 = 1^2 = 1. Then I assume that the...
  31. V

    How to Simplify Trig Identity Proofs?

    More Trig Identity Proofs ... Homework Statement 1. cot^2x - 1 = cot2x ----------- 2cotx2. tanx + cotx = 2csc2x3. cos(A+B) = 1-tanAtanB --------- ---------- cos(A-B) 1+tanAtanB Homework Equations The Attempt at a Solution...
  32. I

    Why are some logical statements not immediately obvious in proofs?

    im just starting to write proofs and it's going well but some things aren't immediately obvious to me. for example it is not immediately obvious to me why \forall_i ~ p_i \vee q_i \Leftrightarrow (\forall_i p_i ) \vee (\forall_i q_i) isn't a tautology and it wasn't immediately obvious...
  33. T

    Proofs for Two Sets of Questions

    Hey guys, I've got two sets of questions here both requiring proofs. Here is a little progress I made with Question Two part c) f=({a,1},{b,1},{c,2},{d,2}) , A={a,c} & B={b,d} f(A)/f(B) = emptyset & f(A/B)={1,2} Any help with the other two parts to Question Two/Three would be great...
  34. B

    Some vector space proofs

    Question 1 Let u, v1,v2 ... vn be vectors in R^{n}. Show that if u is orthogonal to v1,v2 ...vn then u is orthogonal to every vector in span{v1,v2...vn} My attempt if u is orthogonal to v1,v2 ...vn then (u.v1)+(u.v2)+...+(u.vn)=0 Let w be a vector in span{v1,v2...vn} therefore...
  35. B

    Diagonalization & Eigen vectors proofs

    Homework Statement Question 1: A) Show that if A is diagonalizable then A^{T} is also diagonalizable. The Attempt at a Solution We know that A is diagonalizable if it's similar to a diagonal matrix. So A=PDP^{-1} A^{T}=(PDP^{-1})^{T} which gives A^{T}=(P^{-1})^{T}DP^{T} as...
  36. B

    EigenValues & EigenVectors proofs

    Question 1: Proove that if λ is an eigenvalue of [A], then 1/λ is an eigenvalue of [A]{T} Question 2 Proove that a square matrices [A] and [A]T have the same Eigenvalues. Question 3: Show that |det(A)| is the product of the absolute values of the eigenvalues of [A]...
  37. J

    How can I semi-automate mathematical proofs using Mathematica or other tools?

    I am looking for a method to semi-automate mathematical proofs. Precisely, what I would like is that, if for example I define (sorry, I have never used latex in this forum, and I still do not know how to do it yet): f[A] := {y|\existsx\inA (y=f(x))} then, if I have f[A\cupB] what I...
  38. I

    Proving the Existence of Infinite Real Numbers Between Two Given Numbers

    given x<y and x,y,z are elements of R prove there exists at least one z such that x<z<y. proof: x<z<y -> z>x and y>z by the fact that the reals are unbounded there is definitely at least one z such that z>x now either z>y,z<y, or z=y by the order axioms. so... do i just let z<y...
  39. I

    Proving Set Equality and Basic Properties: A Walkthrough

    im just starting to work through vol 1 of apostol and these questions are kind of dumbfounding? #3 on page 15 Let A={1}, B={1,2} Discuss the validity of the following statements (prove the ones that are true) (a) A\subset B (b) A\subseteq B (c) A \in B (d) 1 \in A (e) 1 \subseteq B (f) 1...
  40. O

    Mathematica QED: Uncovering the Meaning Behind Math Proofs

    What does QED stand for behind every mathematical proof? i can't seem to find out what it stands for thanks.
  41. K

    Struggling with Zero Content Proofs?

    http://www.geocities.com/asdfasdf23135/advcal15.JPG Well...I have no idea about this question. I don't even know where to start, and I am having terrible panic on these types of proof. Can someone please explain and guide me through? Believe it or not, my textbook (which is horrible) has...
  42. 1

    Proofs for Triangle Congruency: L is the Midpoint of Line JN

    Homework Statement L is the midpoint of line JN, line PJ congruent line QN, line PL congrent to LINe ql, angel pkj and angle omn are ryte angels. prove: triangle PKJ congruent to TRiangle QMN Homework Equations it mite be line segment, because it has a line on top of it.. no arrows...
  43. Y

    Master Proofs with Ease: Solving Tricky Trigonometric Equations

    Homework Statement prove: sin3x = sinx (3-4sin^2x) tanx+sinx/2tanx = cos^2(x/2) cot2x = (cot^2 x-1)/(2cotx)
  44. Gib Z

    Various Proofs for Irrationality of sqrt2

    We all know the standard proof that the square root of two is irrational, and it's easily extended to all integers that are not perfect squares, but It just striked me yesterday that I have only seen one proof (which really is enough, but still =]). One of the lecturers at the University of...
  45. J

    Proof: f = g + h, Even & Odd Functions

    say function f is continuous on (-\infty,\infty). show that f can be written as f = g + h, where g is an even function and h is an odd function. help pleaseee!
  46. P

    Why do so many professors just do proofs in class?

    All my classes are just profs doing proofs. Great. Too bad the tests requires us to use what we prove to calculate something. For example, my prof spent the first two lectures proving cross and dot product identities (the ones found on the inside covers of many math or physics books). Why...
  47. A

    Experimental proofs of increase of mass with v

    Is there any experiment that shows the increase in mass ( or kinetic energy ) of a moving body when seen from an observer at rest ? I know that sincrotons ( particles accelerators ) must change the frequency bla bla .. But once that these particles have been accelerated and hit a target...
  48. B

    How Do Projections and the Cauchy-Schwarz Inequality Connect in Linear Algebra?

    (a)Let u be a nonzero vector in R^{n}. For all v\epsilonR^{n}, show that proj_{u}(proj_{u}(v)) = proj_{u}(v) and proj_{u}(v - proj_{u}(v)) = \vec{0} (b) An alternate proof of the Cauchy-Schwarz inequality. For v,w \epsilonR^{n}, consider the function q: R -> R defined by q(t) =...
  49. A

    2 Linear Algebra Proofs about Linear Independence

    Homework Statement Proof 1: Show that S= {v1, v2, ... vp} is a linearly independent set iff Ax = 0 has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vector x to the vectors in S 2. The attempt at a solution As far...
  50. Ü

    Can Linear Algebra Proofs Be Mastered with the Right Strategies?

    Homework Statement I'm currently in first year linear algebra... I'm doing quite well, there's just one area of trouble-- proofs. For example: Suppose u.v = u.w, does it follow that v = w? Prove your generalization. Prove that u is orthogonal to v - proju(v) for all vectors u and v in R^n...
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