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. Math Jeans

    Epsilon-Delta Proofs: Math Exam Prep & Book Recommendations

    Hello. I have an upcoming exam for my math course and I am aware that much of it will revolve around Epsilon-Delta proofs. My understanding of them is good enough to prove most limits, but I would be more comfortable being able to answer anything that is thrown at me on this test :confused:. I...
  2. O

    How Do You Prove Trigonometric Identities for Vector Angles?

    First, thanks for all the help so far everyone! vectors a and b exist in the x,y plane and make angles (alpha) and (beta) with x. (Ill use A as alpha and B as beta) prove: cos (A-B) = cos(A)cos(B)+sin(A)sin(B) prove: sin (A-B) = sin(A)cos(B) - cos(A)sin(B) I think there is some...
  3. B

    Proofs on Sets: Help with Proving (A \cup B) X C

    Hello all, I'm having a hard time trying to prove a few things. I'm looking for a little help because I cannot seem to grasp the concept of proofs and what constitutes a valid proof and if my proof is wrong, correcting it. I have a proof done and if anyone could "critique" it I would be...
  4. Saladsamurai

    How Can I Prove Properties of Matrix Multiplication?

    ...involving Matrix Multiplication... I think it is mainly the notation that is killing me here...but it is killing me. Problem: Check parts (2) and (3) of theorem (1.3.18) which says: 1. A(BC)=(AB)C 2. A(B+C)=AB+AC 3. (A+B)C=AC+BC The author led the way on part one with this proof: Let AB=D...
  5. M

    Prove: (S ∩ T) ∪ U = S ∩ (T ∪ U)

    Homework Statement Prove or give a counterexample to each statement. (S ∩ T) ∪ U = S ∩ (T ∪ U) The Attempt at a Solution If I proved by the contrapositive S (T ∩ U) ≠ (S ∩ T) ∪ U where would I go from there. How do I find the contrapositive with the unions and...
  6. B

    Proving B_{r} is Open: Multivariable Proofs

    I have no luck with proofs... Prove that B_{r} ((x_{0}, y_{0})) = {(x,y) : || (x,y) - (x_{0}, y_{0})|| < r} is an open set in R. Now I know that to be an open set if and only if each of its points is an interior point and if it contains no boundary points. I would consider trying to prove...
  7. J

    How Do Proofs for Vector Spaces Over Finite Fields Work?

    It's hard to find the proofs of these theorems. Please help me... Thanks! Theorem 1: Let V be a vector space over GF(q). If dim(V)=k, then V has \frac{1}{k!} \prod^{k-1}_{i=0} (q^{k}-q^{i}) different bases. Theorem 2: Let S be a subset of F^{n}_{q}, then we have dim(<S>)+dim(S^{\bot})=n.
  8. P

    Can a function's continuity be described by a uniform value of delta?

    Can the epsilon associated with f(x) be a function of x? i.e epsilon = delta * (x-2)^2 valid?
  9. A

    Books for Proofs in Limits & Infinite Series Course

    I'm planning on taking a limits and infinite series course soon and was wondering what book(s) I could get off Amazon that would make the process a little less painless when it comes to proofs? Quantifiers, those sort of things, I have no idea about any of it and I'm taking a pre-limits class...
  10. A

    Math Proofs Book: Basics for Beginners

    im looking for a book that shows the fundamentals of proofs cause I am starting and i want to start with the basics. thanks
  11. D

    L1-, L2-, Linfty-Norm Proofs -

    L1-, L2-, Linfty-Norm Proofs - Please Help! Homework Statement Show that ||x||1 < or = n||x||infinity and ||x||2 < or = sqrt(n)*||x||infinity for x exists in the set of all real numbers. Homework Equations ||x||2 is defined here: http://mathworld.wolfram.com/L2-Norm.html ||x||1 is...
  12. K

    Proofs on Limit and Derivatives

    1) Prove that f defined by f(x)= e^(-1/|x|), x=/=0, f(x)= 0, x=0 is differentiable at 0. [I used the definition of derivative f'(0)=lim [f(0+h)-f(0)] / h = lim [e^(-1/|h|) / h] h->0 h->0 and I am stuck here and unable to proceed...] 2) Suppose lim...
  13. L

    How do you get good at proofs?

    I just stare at difficult proofs. I truly do not understand induction. Like if I was to prove Fermat's Little Theorem, I wouldn't know where to start. And I have my final exam next week and I don't know how to study since its all proofs. And if you say do a lot of problems , what happens if I'm...
  14. E

    How Can Beginners Improve Their Mathematical Proof Writing Skills?

    Below is a list of notes on mathematical proofs. The notes are directed at beginners who want to learn how to write mathematical proofs.PROOF TECHNIQUES 1) Introduction to mathematical arguments (by Michael Hutchings) http://math.berkeley.edu/~hutching/teach/113/proofs.pdf 2) How to Write...
  15. MathematicalPhysicist

    Proving Analyticity of Product of Analytic Functions

    i need to prove that if f and and g are analytic functions in (-a,a) then so is fg. well basically i need to find the radius of convergence of fg, which its coefficient is: c_n=\sum_{i=0}^{n} b_i*a_{n-i}, by using cauchy hadamard theorom for finding the radius of convergence, and to show that...
  16. M

    Laurent Series and Singularity Proofs.

    Homework Statement Let D be a subset of C and D is open. Suppose a is in D and f:D\{a} -> C is analytic and injective. Prove the following statements: a) f has in a, a non-essential singularity. b) If f has a pole in a, then it is a pole of order 1. c) If f has a removable singularity...
  17. daniel_i_l

    Proving Matrix Equations: Cramer's Rule, Transpose & Adjoint

    Homework Statement Prove or disprove the following: (A is a nxn square matrix) a) The vector b is in R^n and all its elements are even integers. If all the elements of the A are integers and det(A) = 2, then the equation Ax = b has a solution with only integer elements b) If n is odd and...
  18. W

    Reading proofs - impeding learning?

    When solving a problem, the last thing you want to do is look at the solution. When you're trying to prove a theorem, axiom or whatever, is looking at a proof something that would impede your learning? To me it seems that the answer is yes. Looking at a proof removes the thinking process so...
  19. V

    More Proofs: Prove that if n is an odd positive int., then n^2 = 1(mod 8)

    Homework Statement Prove that if n is an odd positive integer, then n^2\,\equiv\,1\,\left(mod\,8\right). Homework Equations Theorem: a\,\equiv\,b\left(mod\,m\right) if and only if a\,mod\,m\,=\,b\,mod\,m The Attempt at a Solution Using the theorem above: a\,=\,n^2 b\,=\,1,\,m\,=\,8...
  20. JasonRox

    How to find alternative proofs?

    How do I go about finding alternative proofs? I wrote an alternative proof to a theorem including its converse, so I'd like to publish it if it does not yet exist. So far, I just looked into 10 different textbooks that had the theorem. I don't really know much else to do. So far so good...
  21. M

    Proofs of big theorems of calculus

    The following theorems are usually left unproved in calculus, for no good reason. See what you think. 2250: Elementary proofs of big theorems The first theoretical result is the Intermediate Value Theorem (IVT) for continuous functions on an interval. Theorem: If f is continuous on then...
  22. A

    Improving Reading Fluency with Proofs

    Any suggestion on how to improve your reading fluency with proofs of theorems? It's frustrating to spend over 1 hour to read a proof of a theorem that is under 1 page long (or not understanding the proof altogether). Even when every subtopic within a proof is already known, I find that...
  23. L

    Do truth tables accurately prove equality and inequality between sets?

    Are truth tables acceptable forms of proving the equality and inequality between sets. For example, A U (B^C) = (AUB) ^(AUC) A B C AU(B^C) (AUB)^(AUC) F F F F F F F T F F F T F F F F T T T T T F F T T T F T T T T T F T T
  24. R

    Mathematica Mathematical Induction and proofs

    Homework Statement 1. Prove that if n is an even positive integer, then n³-4n is always divisible by 48. 2. Prove taht the square of an odd integer is always of the form 8k+1, where k is an integer. 3. Observe that the last two digits of 7² are 49, the last two digits of 7³ are 43...
  25. R

    Exploring Unique Proofs in Mathematics

    Life is short, and I know I can never experience all of mathematics. So I want to construct a plan to see as many of the unique proofs (across the various disciplines) as possible. (Independently, I'll also proceed to learn as much as possible in depth as well). Reading Munkres'...
  26. G

    Proving Inverse Function of Union Property

    Disclaimer: I might have some problems getting my LaTeX code to work properly so please bear with me while I figure out how to properly use the forum software. Homework Statement The exercise is to prove the following statements. Suppose that f:X \rightarrow Y, the following statement is...
  27. Q

    Need help learning to construct proofs

    so, I am in my first upper level math course beyond required calculus and the introductory linear algebra class. I don't know if it's just a great jump or if I slept through something, but suddenly everything is all about doing proofs. I'm okay with that, and I think it's fabulous because proofs...
  28. A

    Should a General Relativitist Study Math Proofs?

    I know I posted a similar question before but it was moved to the Academic and Career Guidance section and so I got answers from many non-relativists who answered no because they weren't into theoretical physics. So let me be more specific here. Would someone specializing in general...
  29. O

    Proofs of limit laws confusion

    Hey, i was reading through the proof of limit of sum rule in my textbook, and I've ran across somethin i can't understamd. in the proof th textbook uses the triangle inequality: |(f(x) - L) + (g(x)-M)} < e <= |(f(x)-L)|+|(g(x)-M| and then used the latter part in the rest of the...
  30. C

    Geometry Proofs Help - Get Ready For Monday Exam!

    Geometry Proofs.. Help! Please please someone help me! :eek: I have a Geometry Exam on Monday and I don't understand proofs one bit :cry: ! If someone could help me with a few proofs that would be so awesome!
  31. A

    Should a physicist learn math proofs?

    I'm a mathematics specialist with interest in general relativity, and would later like to learn quantum field theory and superstring theory. Of course this requires learning mountains of mathematics that I haven't even learned yet because I spend 80% of my studies doing math proofs. Doing...
  32. A

    Will computers be able to do math proofs?

    I'm a math graduate student, but also have great interest in computer capabilities. Chess players once thought that a machine cannot beat the world's best chess players because they cannot plan like humans despite their calculational power. Nowadays, computers are consistently beating the...
  33. Schrodinger's Dog

    Are proofs essential in understanding mathematics in physics?

    I've noticed a lot of mathematics on this forum revovles around proofs, but have not really come across any in depth proofs as such and so am unfamilliar with much of the topics discussed here, and in fact some I can't really follow. I got to thinking though at which point does it become...
  34. G

    Epsilon-Delta Proofs of Limits

    Trying to press on through Epsilon-Delta proofs of limits (for more than one variable) and yet there's only one example I've found thus far of even a multi-variable Epsilon-Delta proof. Would it be possible for someone to solve the Epsilon-Delta proof of the limit: (xy^2)/(x^2+y^2). Note...
  35. B

    Examples of proofs involving geometrical forms

    I am looking for some good websites that have proofs involving parallelograms and rhombus'? preferably in statement and reasons format any help would be appreciated. thank you
  36. K

    QM having difficulty on proofs of operators

    I know this is a simple part of Quantum Mechanics, but I seem to be having trouble with it, I'm not sure if my math is just wrong or if I'm applying it wrong. The questions that I have are: Prove the following for arbitrary operators A,B and C: (hint-no tricks, just write them out in...
  37. W

    Really want to study formal math with proofs

    I was wondering i really want to study formal math with proofs etc. Are there any good books out there?
  38. S

    Calculus Proofs Help: Get Answers Now!

    Hey, I need help with a couple of questions in my analysis calc and proof class. Thanks in advance! Prove that S = { n−1 |n ∈ N} is bounded above and that its supremumn is equal to 1. Use the Intermediate Value Theorem to show that the polynomial x4 + x3 − 9 has at least two...
  39. M

    Is Every Group with Only Cyclic Subgroups Itself Cyclic?

    I m having trouble with a couple group theory proofs. I just have no clue how to start. If u could put me on the right path that would be great. first prove of disprove that if every subgroup of a group G is cyclic, then G is cyclic. and second prove or disprove that every group X of...
  40. A

    Proving Linear Algebra Concepts: Rank, RREF, Invertibility, and Dependency

    1) Find two matrices A and B where Rank [AB]≠Rank(BA) 2) Find a matrix A where Rref(A)≠Rref(A^T) where T is the transpose 3) Find X given that B is invertible if BXB^-1 –A=I_n (identity matrix) 4) Prove that [Ab_1 Ab_2 Ab_3] is linearly dependent given that {b1 b2 b3} is linearly...
  41. P

    Programs Do physics majors need to know math proofs?

    I'm a current physics major and considering whether I should minor or double major in math as well. Since I've heard that physics majors need to take upper-division linear algebra and analysis, and I'm currently taking a mulitvariable calculus course, should I be spending time trying to...
  42. T

    Proofs of Logarithms: Proving Equality of a and b

    Can't start: (log_{a}b)(log_{b}a) =1
  43. C

    Can a Matrix with Identical Columns be Invertible?

    i need to be able to prove that an nxn matrix with two identical columns cannot be invertible. I know that if the columns of the matrix are linearly independent then the matrix is invertible. Could some please give me a hint on how to do this proof because i really don't know where to start...
  44. D

    Epsilon-Delta Proofs: Understanding the Process

    Hi, Why is it, that when ever epsilon-delta proofs are done, once delta is found in terms of epsilon, it is reinputed through again? Is there any point to this really?
  45. A

    Are Multiple Substitutions Allowed in Proofs?

    I'm reading Introduction to Mathematical Logic gy by Vilnis Detlovs and Karlis Podnieks, and I'm confused about proofs. In the book, it says that to prove directly you should find ways to substitute the hypoethesis formula(s) into one of the axiom schemas so that other formulas will be...
  46. M

    How Do Parallel Line Proofs Determine Interior Angles in a Triangle?

    I need to prove that <acb is equal to one of the other interior angles of triangle abc. help when pic uploads
  47. michael879

    Unravelling the Mystery of Light's Constant Speed: Challenges and Proofs

    I get the theory of special relativity, it is the logical conclusion drawn from the two facts that: a) the laws of physics are the same in all reference frames b) the speed of light is constant in all reference frames what I don't get is why einstein thought the speed of light was constant...
  48. M

    Proving Squares of Odd Integers Always 8k+1

    Here is the question. I have to prove it. Prove that the square of an odd integer is always of the form 8k+1, which k is an integer. Now I do not know how to start it. But this is what I came up with. odd integer= 2k+1 therefore the square of an odd integer (2k+1)^2 i have used...
  49. Reshma

    How does the kinetic energy change for a varying mass?

    Show that for a single particle with constant mass the equation of motion implies the following differential equation for the kinetic energy: {dT\over dt} = \vec F \cdot \vec v while if the mass varies with time the corresponding equation is {d(mT)\over dt} = \vec F \cdot \vec p Proof...
  50. C

    Proofs of Irrationality Correct?

    Are 2b, 2c, and 2d correct? The last part of 2d I am getting stuck. http://www.artofproblemsolving.com/Forum/weblog.php?w=564 note: you can comment on the site as a guest Thanks
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