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. B3NR4Y

    Simple Set Theory Proofs: A Guide to Proving Set Identities - Homework Help

    Homework Statement 1. Prove that if A \cap B = A and A \cup B = A , then A = B 2. Show that in general (A-B) \cup B \neq A 3. Prove that (A-B) \cap C = (A \cap C) - (B \cap C) 4. Prove that \cup_{\alpha} A_{\alpha} - \cup_{\alpha} B_{\alpha} \subset \cup_{\alpha} (A_{\alpha} -...
  2. D

    Proving properties of the Dirac delta function

    I've been thinking about the properties of the Dirac delta function recently, and having been trying to prove them. I'm not a pure mathematician but come from a physics background, so the following aren't rigorous to the extent of a full proof, but are they correct enough? First I aim to...
  3. S

    MHB [Limits] Help with Delta-Epsilon Proofs for Multivariable Functions

    Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions. here is my question: Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists. The part that has me confused is the y to the power of 3, where...
  4. D

    MHB Fundamental theorem and limit proofs

    Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero. The hint is to use Stirling's approximation. What is this?
  5. SpiderET

    Lorenz covariance vs proofs of relativity theory

    I have been studying history of relativity theory and now it seems to me, that it is wrong to automatically assume that proofs of Lorentz covariance are proofs of Special relativity theory. It seems to me, that there is broader group of theories, that are compatible with Lorentz covariance but...
  6. D

    Proving the reciprocal relation between partial derivatives

    If three variables x,y and z are related via some condition that can be expressed as $$F(x,y,z)=constant$$ then the partial derivatives of the functions are reciprocal, e.g. $$\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$$ Is the correct way to prove this the following...
  7. S

    MHB Proofs of Modular Division

    Let $a$, $b$, and $c$ be integers, where a $\ne$ 0. Then $$ $$ (i) if $a$ | $b$ and $a$ | $c$, then $a$ | ($b+c$) $$ $$ (ii) if $a$ | $b$ and $a$|$bc$ for all integers $c$; $$ $$ (iii) if $a$ |$b$ and $b$|$c$, then $a$|$c$. **Prove that if $a$|$b$ and $b$|$c$ then $a$|$c$ using a column proof...
  8. F

    What is the theorem with the most proofs?

    I wonder which theorem has the most proofs, or has been proven in the most ways? I know of Loomis' The Pythagorean Proposition which came out decades ago & contains 370 proofs & more, & the proofs are even catalogued into four types (algebraic, geometric, etc). So that makes me think the...
  9. N

    Finding Delta for a Given Epsilon and Limit: 3-2x, x0=3, E=.02

    Given a function f(x), a point x0, and a positive number E (epsilon), write the limit then find delta>0 such that for all x 0< |x-x0| < delta -> |f(x)-L| < E f(x) = 3-2x, x0=3, E=.02 Here is my attempt: Lim (3-2x) as x->3 = -3 -.02 < |3-2x - 3| <.02 -.02 < |-2x| < .02 .01 > x > -.01 -2.99 > x-3...
  10. D

    Slight confusion in proof of Hadamard's Lemma

    I've been reading Wald's book on General Relativity and in chapter 3 he introduces and uses the so-called Hadamard's Lemma: For any smooth (i.e. C^{\infty}) function F: \mathbb{R}^{n}\rightarrow\mathbb{R} and any a=(a^{1},\ldots,a^{n})\in\mathbb{R}^{n} there exist C^{\infty} functions H_{\mu}...
  11. D

    Addition property of integration intervals proof

    First of all, apologies as I've asked this question before a while ago, but I never felt the issue got resolved on that thread. Is it valid to prove that \int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx using the fundamental theorem of calculus (FTC)?! That is, would it be valid to do...
  12. C

    Need some guidance on Linear Algebra proofs

    First, let me say that I am a senior physics undergrad. I have failed Linear Algebra once before. Otherwise I am a straight A student. I'm also taking Ordinary Differential Equations right now, and I breeze through that class without a care in the world. I'm not sure if I've developed some sort...
  13. S

    How to Choose the Right Proof Method for Mathematical Propositions?

    Homework Statement Prove the following propositions: 1) ∀x ∈ (0, 1), ∃y ∈ (0, 1), x < y and 2) ∀x, y ∈ R, if x < y, then ∀b ∈ (0, ∞), ∃a ∈ (0, ∞), x + ab < y. Can anyone help me out with either one? I have a few others that I can get but I can't get these two. Mainly because these don't...
  14. C

    Show that for each a < b a, b ∈ N we have the following

    1) 3^(2^a) + 1 divides 3^(2^b) -1 2) If d > 2, d ∈ N, then d does not divide both 3^(2^a) + 1 and 3^(2^b) -1 Attempt: Set b = s+a for s ∈ N m = 3^(2^a). Then 3^(2^b) - 1 = 3^[(2^a)(2^s)]-1 = m^(2^s) -1 Thus, m+1 and m-1 divides m^(2^s) -1 by induction. If s = 1, then m^(2^s) -1 = m^2 -...
  15. J

    Proofs involving negations and conditionals

    0. Background First and foremost, this is a proof-reading request. I'm going through Velleman's "How To Prove It" because I found that writing and understanding proofs is a prerequisite to serious study of mathematics that I did not meet. Unfortunately, the book is very light on answers to its...
  16. Porthos

    Which is more useful for a physics degree: logic or proofs?

    I know that I have already posted a couple of threads like this one (albeit dealing with different courses), but I have had excellent responses here, and I was hoping I could get a couple more. I know that it is best to speak to academic advisors and professors, but there are only a couple that...
  17. M

    Is learning epsilon-delta proofs before analysis a good idea

    Hello PF people. It's my first post here, but I have been lurking around this forum for awhile now. I'm currently learning differential calculus using a text by Stewart and I want to attain a better comprehension of pure mathematics. My question is: would it be a good idea to get another text...
  18. D

    Irreducible linear operator is cyclic

    I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...
  19. U

    Ross Elementary Analysis Epsilon Delta Proofs

    Does Ross's book teach and/or use Epsilon-delta proof techniques?
  20. U

    Calculus Spivak Calculus on Manifolds and Epsilon delta proofs

    I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...
  21. dkotschessaa

    I need a better "method" for working out proofs and problems

    I will even take book recommendations, though I have read Polya's "how to solve it," and Vellemans similarly titled "How to Prove it." I think I am looking more for how to organize my thoughts, and much of this overlaps with "how to study," which, I am still trying to learn how to do. My...
  22. J

    Analyzing Logical Arguments: Not A, B or Not C, B→ (A and D), E→(C)

    Homework Statement Hypotheses: not a, b or not c, b→ (a and d), e→(c) Conclusion: not e 2. The attempt at a solution: So far, I have this: 1) not a as premise 2) b or not c as premise 3) b→ (a and d) as premise 4) e→(c) as premise 5) a by Step 1 and Law of Excluded Middle. 6) c is true...
  23. C

    Solve the recurrence relation using iteration

    Homework Statement [/B] Solve the recurrence relation (use iteration). an = an-1 + 1 + 2n-1 a0 = 0 Then prove the solution by mathematical induction. Homework EquationsThe Attempt at a Solution a1 = 2 a2 = 5 a3 = 10 a4 = 19 a5 = 36 The solution appears to be an = n + 2n - 1 How are we...
  24. A

    MHB Considering the criterion for limit proofs

    In a proof. Prove that **given**: $$\lim_{x \to a} f(x) = L$$ then $$\lim_{x\to a} |f(x)| = |L|$$ We know that $$|f(x) - L| < \epsilon \space \text{for} \space |x - a| < \delta_1$$ What is the objective then? Do we prove there exists a $\delta_2$ such that $\displaystyle \lim_{x\to a}...
  25. H

    Vacuous "If then" statements: Can you use direct proofs?

    Hi, I'm comfortable using a direct proof to prove ##P → Q## type statements when I have a ##P## that is either always true (e.g ##x=x##) or can be true (e.g. ##x > 3##). But what about when ##P## is definitely false, (e.g. ##x \neq x##), or definitely false in relation to an earlier statement...
  26. R

    Question regarding proofs and theorems

    Hey guys, I have been interested in formalistic mathematics for a while, about a year now. Every time I read a formalistic book on math (Principles of Mathematical Analysis by Rudin is a great example) I never understand how mathematicians develop the structure they present in the books. And...
  27. A

    Can anybody check this proof for a Sine limit?

    Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets). I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space). Homework Statement Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0 Homework Equations Given below: The...
  28. topsquark

    MHB Induction Proofs & Negative Proofs: Examining POTW 135

    This is in reference to a POTW, http://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-135-october-27th-2014-a-12786.html. The logic behind this problem is simple, the number 2^{2^x} can only have factors of 2. But (n + 1)^3 - 1 contains an odd factor. Great...
  29. T

    Proving vectors are in the column space

    How would you prove that adding two vectors in the column space would result in another vector in the column space? I know this is maybe the most basic property of vectors and subspaces, and that the very definition of the column space says it's spanned by vectors in the column space. Is there...
  30. G

    Sequence (n)/(n^n) Convergent or Divergent and Limit?

    Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...
  31. A

    MHB Algebraic Proofs and Verifying identities

    Hey all! I am having some trouble with a certain problem on my homework. I would like some guidance. I have to prove one side of the equation is equal to the other, as you may know, as this is an algebraic proof. This in itself isn't too hard. The hard part is just this one particular problem. I...
  32. C

    Proving f(x): One-to-One, Onto, or Both?

    Homework Statement Prove whether the function f(x) = x/(1+x^2) with domain & codomain = reals is one-to-one, onto, or both. Homework EquationsThe Attempt at a Solution I know to show if it's one-to-one I have to show a/(1+a^2) = b/(1+b^2), ultimately that a = b, I don't know how to simplify...
  33. B

    One Physics MS semester left - what courses to take?

    Hello all, I've got one more semester before I earn my physics MS, and I have space for one or two extra courses. I am going into oceanography, and I would like to have a strong foundation in math in order to understand the theory I'll encounter as well as possible. Lots of physical...
  34. I

    MHB Solving Limits with Delta-Epsilon Proofs

    PLEASE HELP! i am so lost on this. we're using delta epsilon proofs and i am so confused since it was never properly taught to me in calc 1. find the limit. $\lim_{{(x,y)}\to{(0,0)}}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$
  35. Q

    Why do I just not "get" math proofs?

    The only proof-based math class I've taken so far was on abstract algebra. Concepts were easy for me to understand, but I was constantly having trouble with some of the proofs. I so frequently get this feeling that the last, tiny trivial step left in my proof is just "right there," and yet I...
  36. 1

    Easier to Grade Proofs: A vs. B

    To those of you who may grade proofs in which the result is not stated (ie prove or disprove), which of the following do you think is easier to grade, and a better format: A) Proof... Therefore the theorem is false/true. B) The theorem is false/true. To see this, consider the following...
  37. I

    Problems with proofs of Robert Geroch mathematical physics

    Hello guys, I'm new in this forum, this is my first Thread. I've started reading Robert Geroch's Mathematical Physics recently and I've been having problems with some of the proofs that involve monomorphism. He defines monomorphism the following way (pg 4): let ψ be a morphism between A...
  38. A

    MHB What is the Domain for the Inverse of a One-to-One Function?

    Hey guys, I've a few more questions this time around from my problem set: (Ignore question 2abc, I only need help with the first one) Question: For the first one, in order to prove that a function is one-to-one, f(x1) =/ f(x2) when x1 =/ x2. Thus, the horizontal test applies. So I said...
  39. S

    Set theory: proofs regarding power sets

    Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X). I really don't know how to get started on this proof but I tried to start with something like this: ∀ m,n ∈ A,B ⇒ m,n ∈ X ⇒ Is this the right way to start on this proof...
  40. D

    Mathematical proofs and an understanding of them for scientists

    To what extent, if any, is an understanding of mathematical proofs required for a scientist? I can empathize with a need for an understanding of the general machinery of the tools you are using (understanding, for example, how it is the chain rule came about, ie, how it was derived) but, using...
  41. Amcote

    Elementary Number Theory - GCD problems and proofs

    Problem 1 Suppose ab=cd, where a, b, c d \in N. Prove that a^{2}+b^{2}+c^{2}+d^{2} is composite. Attempt ab=cd suggests that a=xy, b=zt, c=xz. d=yt. xyzt=xzyt. So (xy)^{2}+(zt)^{2}+(xz)^{2}+(yt)^{2}=x^{2}(y^{2}+z^{2})+t^{2}(z^{2}+y^{2})=(x^{2}+t^{2})(z^{2}+y^{2}) Therefore this is...
  42. P

    Calculus Proofs Help Thanks

    Hi, I've been trying a couple of proofs that my calc teacher gave me, but I'm not sure if I have the right approach or not. 1) Prove that the degree of the depressed polynomial is exactly one less than the degree of the original polynomial. - For this proof, all I can come up is the face...
  43. M

    Algebraic proofs of divisibility

    Hello, I have a problem with algebra and divisibility etc. I have a swedish textbook that really sucks. Not a good solutions section and no separate solutions manual either. Just a lot of proofs to show. At the moment I'm stuck at proofs with divisibility. I have two examples: 1)...
  44. nougiecat

    Kochen-Specker Proofs Look Wrong to Me

    Can someone explain to me what is wrong with the following argument? There are two parts. First of all, K-S, despite passing reference to hidden variables, doesn't really seem to depend on any interesting properties of HV, but instead appears to be an indictment of QM itself by asserting that QM...
  45. M

    Discrete proofs involving divisibility

    I'm trying to do some extra course work to prepare for my final next week but I'm having a lot of trouble with the book problems. They talk about a lot of things we weren't taught. Can someone help me out here? Prove: n\niZ, n= a multiple of gcd(a,b) ⇔ n is a linear combination of a and b This...
  46. S

    Proofs by contraposition and contradiction

    Homework Statement (a) Prove that if n is an integer and n2 is a multiple of 3, then n is a multiple of 3. (b) Consider a class of n students. In an exam, the class average is k points. Prove, using contradiction, that at least one student must have received at least k marks in the exam...
  47. M

    Proofs with current density and wavefunctions

    Homework Statement So I was able to find a problem that was kind of similar to a homework problem that I am working on. Unfortunately, I'm not quite sure what is going on partially within the problem. In the problem they state that \phi=\phi*, but it does not state why. I was wondering...
  48. P

    Clarifications on some types of proofs

    Homework Statement So I would know how to prove a statement like \sqrt{2} by contradiction, all you have to do is assume to negation. But what about something like p → q Like if p = (bc mod a != 0), q = (b mod a != 0), how would I prove this, would I negate q or p, or both?
  49. P

    Proving Conclusion of R: Using Rules of Inference and Given Premises

    Homework Statement (p \wedge t)\rightarrow (r \vee s),q \rightarrow (u \wedge t), u \rightarrow p, \neg s, q, show that these premises imply the conclusion of rThe Attempt at a Solution The question calls for rules for inference to solve this problem, how would I go about doing that...
  50. M

    Gauss Law: Proofs and Electromagnetism Resources

    about gauss law! is there any precise proof for gauss law? why ø=q/ε always regardless how the charge is distributed inside the surface and if anyone know a good book for electromagnetism please type its name for me.. thank u
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