What is Proof: Definition and 999 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. koustav

    Are Spacelike and Timelike Orthogonal: Mathematical Proof Explained

    are spacelike and timelike orthogonal?what is the mathematical proof
  2. J

    MHB Proving an Integral with a Direct Proof & Epsilon Argument

    Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...
  3. S

    A Proof - gauge transformation of yang mills field strength

    In Yang-Mills theory, the gauge transformations $$\psi \to (1 \pm i\theta^{a}T^{a}_{\bf R})\psi$$ and $$A^{a}_{\mu} \to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$ induce the gauge transformation$$F_{\mu\nu}^{a} \to F_{\mu\nu}^{a} -...
  4. J

    MHB Real Analysis - Riemann Integral Proof

    I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?
  5. P

    Explosion and conservation of momentum problem

    Note: Please only give hints please! No answers because I want the satisfaction of solving it. 1. Homework Statement A mass M at height h above flat round and falling vertically with velocity v breaks up explosively into 2 parts. The kinetic energy given to the system in the explosion is E...
  6. T

    Is S1 Always a Subset of S2 If R1 Is a Subset of R2?

    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...
  7. DaniV

    I Does the Tail of a Convergent Series Also Converge to Zero?

    {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N}\sum_{n=N+1}^{\infty}an is also converage proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0 {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N} \sum_{n=N+1}^{\infty}an is...
  8. B

    Trying to understand a proof about ##\lim S##

    Homework Statement I am trying to understand the proof that ##\lim S## is a closed set in the metric space ##M##, where ##\lim S = \{ p \in M ~|~ p \mbox{ is a limit point of } S\}##. Here is the definition of a limit point: ##p## is a limit point of ##S## if and only if there exists a...
  9. lfdahl

    MHB Proving Inequality for Variables with Constraints

    Let $0 \le a,b,c \le 1.$ Prove the inequality:$\sqrt{a(1-b)(1-c)}+ \sqrt{b(1-a)(1-c)}+\sqrt{c(1-a)(1-b)} \le 1 + \sqrt{abc}$
  10. F

    MHB Proof of Knaster-Tarski Theorem

    Let $F:P(A)->P(A$) be monotone and $C$ be the union of sets whose image is invariant under F. Prove $F(C)=C$ https://i.stack.imgur.com/3Wjdg.png
  11. F

    MHB How to Prove a Set Theory Ordinal Relationship?

    Let $\beta$ be an ordinal. Prove that $A\cap \bigcup\beta=\bigcup\{A\cap X\mid X \in \beta\}$ I'm not sure on this. It looks a bit like union distributing over intersection. Please help.
  12. M

    Proof by induction, ##(n)^{2} \le (2n)##.

    Homework Statement I need to prove by induction that ##(n!)^{2} \le (2n)!##. I'm pretty sure about my preliminary work, but I just need some suggestions for the end. Homework Equations It is well known from a theorem that if ##a \le b## and ##c \ge 0##, then ##ca \le cb##. The Attempt at a...
  13. A

    I Regarding Cantor's diagonal proof

    I am very open minded and I would fully trust in Cantor's diagonal proof yet this question is the one that keeps holding me back. My question is the following: In any given infinite set, there exist a certain cardinality within that set, this cardinality can be holded as a list. When you change...
  14. M

    Ε-δ proof: lim x->a f(x) = lim h->0 f(a + h)

    This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate. Homework Statement Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h) Homework EquationsThe Attempt at a Solution By the...
  15. 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...
  16. D

    I Proof that parity operator is Hermitian in 3-D

    Hi. I have been looking at the proof that the parity operator is hermitian in 3-D in the QM book by Zettili and I am confused by the following step ∫ d3r φ*(r) ψ(-r) = ∫ d3r φ*(-r) ψ(r) I realize that the variable has been changed from r to -r. In 3-D x,y,z this is achieved by taking the...
  17. Adgorn

    Proof regarding determinant of block matrices

    Homework Statement Let A,B,C,D be commuting n-square matrices. Consider the 2n-square block matrix ##M= \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}##. Prove that ##\left | M \right |=\left | A \right |\left | D \right |-\left | B \right |\left | C \right |##. Show that the result may not be...
  18. Yiming Xu

    I Express power sums in terms of elementary symmetric function

    The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...
  19. binbagsss

    Elliptic functions proof -- convergence series on lattice

    Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...
  20. M

    I For direct proof, how do you choose M for bounded sequence?

    So the definition of a bounded sequence is this: A sequence ##(x_{n})## of real numbers is bounded if there exists a real number ##M>0## such that ##|x_{n}|\le M## for each ##n## My question is pretty simple. How does one choose the M, based on the sequence in order to arrive at the...
  21. A

    I When is D_{n} abelian? What's wrong with the proof?

    I agree that this could have been done more simply(i'm not looking for an alternative proof), but I don't understand how it is wrong, any insight? Since Dn is an dihedral group, we know its elements are symmetries, Dn = (R1,R2,R3...Ri) and since R is a symmetry, we know it's a permutation, so...
  22. UsableThought

    Help with algebraic deduction steps in a proof by induction

    I'm in the 6th week of a well-known MOOC course created by Kevin Devlin, "Introduction to Mathematical Thinking." I enjoy the course & did well in the first weeks with conditionals and truth tables, etc.; however now that we are entering into proofs, I'm running into trouble with algebra...
  23. J

    MHB Show How to Prove $\binom{n}{r}$ with Pascal's Triangle

    Repeatedly apply $\binom{n}{r}= \binom{n-1}{r}+\binom{n-1}{r-1}$ to show: $$\binom{n}{r}=\sum_{i=1}^{r+1}\binom{n-i}{r-i+1}$$ The closest i got was showing you could show different iterations with the binomial coefficients (Pascal's Triangle).
  24. binbagsss

    Elliptic functions proof - finitely many zeros and poles

    Homework Statement Hi I have questions on the attached lemma and proof. ##f(z)## is an elliptic function here, and non-consant ##\Omega## is a period lattice. So the idea behind the proof is this is a contradiction because the function was assumed to be non-constant but by the theorem that...
  25. F

    I Proof that lattice points can't form an equilateral triangle

    From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...
  26. chwala

    How can we prove ##e^{ln x}= x## and ##e^-{ln(x+1)}= \frac 1 {x+1}##?

    Homework Statement they say 1. ##e^{ln x}= x ## and 2.##e^-{ln(x+1)}= \frac 1 {x+1}## how can we prove this ##e^{ln x}= x ## and also ##e^-{ln(x+1)}= \frac 1 {x+1}##? Homework EquationsThe Attempt at a Solution let ## ln x = a## then ##e^a= x, ## a ln e= x,## →a= x, where ## ln x= x
  27. Math Amateur

    MHB Exploring Proposition 6.1.7 and its Proof in Bland's "Rings and Their Modules"

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ... Proposition 6.1.7 and its proof read as follows: In the above text from Bland, in the proof of (1), we read the...
  28. L

    MHB Getting Nowhere with a Proof Question: Help Needed

    I'm stuck on this proof question: (¬(Q⇒¬P) ∧ ¬((Q∧¬R)⇒¬P )) ⇔ ¬(R ∨ (P ⇒¬Q)) I've tried to get rid of the negation and implications but I keep going in circles and I'm getting nowhere near to the equivalence required. I would appreciative if anyone can help me solve this because it's really...
  29. Duke Le

    Where is wrong in this proof for rotational inertia ?

    Homework Statement Prove the formula for inertia of a ring (2D circle) about its central axis. Homework Equations I = MR^2 Where: M: total mass of the ring R: radius of the ring The Attempt at a Solution - So I need to prove the formula above. - First, I divide the ring into 4...
  30. T

    Proof of the total probability rule for expected value?

    Homework Statement Does anyone know of a simple proof for this: https://s30.postimg.org/tw9cjym9t/expect.png E(X) = E(X|S)P(S) + E(X|S_c)P(S_c) X is a random variable, S is an a scenario that affects the likelihood of X. So P(S) is the probability of the scenario occurring and and P(S_c) is...
  31. B

    A Residue Proof of Fourier's Theorem Dirichlet Conditions

    Whittaker (1st Edition, 1902) P.132, gives two proofs of Fourier's theorem, assuming Dirichlet's conditions. One proof is Dirichlet's proof, which involves directly summing the partial sums, is found in many books. The other proof is an absolutely stunning proof of Fourier's theorem in terms of...
  32. Mr Davis 97

    Linear Independence of a Set of Vectors

    Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...
  33. binbagsss

    Q about the proof of periods of non-constant meromorphic functions

    Homework Statement [/B] Theorem attached. I know the theorem holds for a discrete subgroup of ##C## more generally, ##C## the complex plane, and that the set of periods of a non-constant meromorphic function are a discrete subset. I have a question on part of the proof (showing the second...
  34. A

    Vector Proof Homework: The Rotation Matrix

    Homework Statement Homework Equations The Rotation Matrix The Attempt at a Solution I am sorry but I do not know how to even begin
  35. X

    (Number theory) Sum of three squares solution proof

    Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...
  36. L

    Proof Check: Geometry AB=EF If A=/B

    Homework Statement Let A and B be elements of the line EF such that A=/B prove that the line AB=EFHomework Equations Axiom that two points determine a unique line and that the intersection of two lines has two distinct points then these lines are the same. The Attempt at a Solution [/B] If A...
  37. binbagsss

    I Proving Hurwitz Identity: Modular Forms & Beyond

    Hi, My notes say that hurwitz identity currently has no elementary proof? One way to prove the identity is through modular forms: to consider Eisenstein series, ##E_4^2## and ##E_8## , note that the dimension of space of modular functions of weight 8 is one, find the constant of...
  38. binbagsss

    Q about a Proof -- periods meromorphic function form discrete set

    Homework Statement Hi, As part of the proof that : the set of periods ##\Omega_f ## of periods of a meromorphic ##f: U \to \hat{C} ##, ##U## an open set and ##\hat{C}=C \cup \infty ##, ##C## the complex plane, form a discrete set of ##C## when ##f## is a non-constant a step taken in the...
  39. binbagsss

    Sin inequality proof , ##0 \leq 2x/\pi \leq sin x##

    Homework Statement Homework EquationsThe Attempt at a Solution Hi How do I go about showing ##0 \leq \frac{2x}{\pi} \leq sin x ##? for ## 0 \leq x \leq \pi /2 ## I am completely stuck where to start. Many thanks. (I see it is a step in the proof of Jordan's lemma, but I'm not interested in...
  40. T

    Using symbolic logic in mathematical proof?

    is this a practical way of proving math theorems? i asked because when i tried, it seemed difficult for me to decide as to how exactly i should translate theorems and given statements into logical forms and since there are so many different ways, i do not know which one is correct. For example...
  41. VrhoZna

    Proof regarding direct sum of the dual space of a v-space

    (From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V. 1. Homework Statement Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let $$V_j = W_1 + \cdots + W_{j-1}...
  42. S

    I Why isn't tunnelling considered proof of hidden variables?

    When I hear that mass of a particle has managed to hop through a solid barrier ..it tells me that the mass was a variable and not physical at the time.
  43. L

    I How can I improve my proof skills with internet resources?

    Is this proof even correct?! It places assumption on a and c NOT BEING ZERO. Thanks in advance. I am new to proofs.
  44. evinda

    MHB Questions about proof of theorem

    Hello! (Wave) We say that the space $\Omega$ satisfies the exterior sphere condition at the point $x_0 \in \partial{\Omega}$ if there is a $y \notin \overline{\Omega}$ and a number $R>0$ such that $\overline{\Omega} \cap \overline{B_y(R)}=\{ x_0 \}$. Let the function $\phi \in...
  45. B

    Verifying a Proof about Maximal Subgroups of Cyclic Groups

    Homework Statement Show that if ##G = \langle x \rangle## is a cyclic group of order ##n \ge 1##, then a subgroup ##H## is maximal; if and only if ##H = \langle x^p \rangle## for some prime ##p## dividing ##n## Homework Equations A subgroup ##H## is called maximal if ##H \neq G## and the only...
  46. I

    How Do You Prove a Function is Not Uniformly Continuous?

    Homework Statement Let ##f:X \to Y##. Show that ##f## not uniform continuous on ##X## ##\Longleftrightarrow## ##\exists \epsilon > 0## and sequences ##(p_n), (q_n)## in ##X## so that ##d_X(p_n,q_n)\to 0 ## while ##d_Y(f(p_n),f(q_n))\ge \epsilon##. Homework Equations Let ##f:X\to Y##. We say...
  47. I

    Is a Complete Subspace Necessarily Closed in a Metric Space?

    Homework Statement Let ##E## be a metric subspace to ##M##. Show that ##E## is closed in ##M## if ##E## is complete. Show the converse if ##M## is complete. Homework Equations A set ##E## is closed if every limit point is part of ##E##. We denote the set of all limit points ##E'##. A point...
  48. I

    Convergence of sequence in metric space proof

    Homework Statement Let ##E \subseteq M##, where ##M## is a metric space. Show that ##p\in \overline E = E\cup E' \Longleftrightarrow## there exists a sequence ##(p_n)## in ##E## that converges to ##p##. ##E'## is the set of limit points to ##E## and hence ##\overline E## is the closure of...
  49. CynicusRex

    Prove that if a² + ab + b² = 0 then a = 0 and b = 0

    Homework Statement Prove that if a² + ab + b² = 0 then a = 0 and b = 0 Hint: Recall the factorization of a³-b³. (Another solution will be discussed later when speaking about quadratic equations.) Homework Equations a² + ab + b² is close to a² + 2ab + b² = (a+b)² a³-b³=(a-b)(a²+ab+b²) The...
  50. mastrofoffi

    Proof of Dirichlet stability theorem

    Hola, I tried to give a proof of this theorem and then check it against the one given by my book(Fasano, Marmi - Analytical Mechanics); I feel like mine seems reasonable and pretty intuitive, but the one on the book is a bit different and I don't really understand it completely, so I'd like to...
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