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. E

    Self study Multivariable Calculus or Introduction to Proofs?

    Hello. I was wondering if I should self study multivariable calculus or introduction to proofs? I am an entering high school senior (contrary to what my username might suggest), and I just took a Calc 2 class last spring. I can only do one or the other, and I don't know which one would be...
  2. A

    Math software for checking solutions and proofs?

    Hello guys, I don't know where else to post this but here goes. I'm going to be catching up on a looot of math this year. Unfortunately a lot of the math books that I'll be using only provide the answers to odd numbered questions. And the answers that they do provide a lot of the times "do...
  3. S

    Proving Basic Set Theory: Trichotomy, Union, Intersection, and Multiplication

    At first glance these things seem so intuitive and familiar from other maths (like distribution) that I don't see how/where to start in proving them; while I know its probably quite simple. I understand what union and intersection are, but I'm unsure if multiplying two sets means a new set with...
  4. Z

    Delta-Epsilon Proofs: Finding d for f(x)+g(x)=6

    Homework Statement Suppose the functions f and g have the following property: for all E > 0 and all x, if 0 < |x - 2| < sin((E^2)/9) + E, then |f(x) - 2| < E, if 0 < |x - 2| < E^2, then |g(x) - 4| < E. For each E > 0, find a d > 0 such that, for all x, i) if 0 < |x - 2| < d, then...
  5. X

    I really do not get proofs AT ALL.

    I really do not get proofs AT ALL. Stuff like this... "Prove that (n+1)2 \geq3n if n is a positive integer with n\leq4." Proof by exhaustion would be applied here.. what the book tells me. "Show that there are no solutions in integers x and y of x2+3y2=8." Then there's also...
  6. Z

    Prove Irrationality of \sqrt{3}, \sqrt{5}, \sqrt{6} and 2^1/3, 3^1/3

    Homework Statement a) Prove that \sqrt{3}, \sqrt{5}, \sqrt{6} are irrational. Hint: To treat \sqrt{3}, for example, use the fact that every integer is of the form 3n or 3n + 1 or 3n + 2. Why doesn't this proof work for \sqrt{4}? b) Prove that 2 ^ 1/3 and 3 ^ 1/3 are irrational. Homework...
  7. D

    Courses Pass math course with proofs by memorization?

    I'm studying applied physics and I am currently in my second semester of the second year. I now have a probability and mathematical statistics course which is causing me a problem. Although I had lots of math prior to this course, none of it actually required writing proofs. Yet the spring...
  8. J

    Math Proving: Are Computer Derived Theorems Accurate?

    If we feed all the existing mathematical axioms to a powerful computer, it should be able to give us all the proofs and theorems that can be derived using the axioms. Is there anything wrong with this logic?
  9. S

    Proving EM Doppler Shift Ratio: v/c = (r^2 - 1) / (r^2 +1)

    Hi there, I have some exams later this month, and some of the previous exam questions are to prove a formula given another formula fx here with EM doppler shift: define ratio: r= f/ f0 using relativistic doppler frequency for EM: f = square root of: ((c+v) / (c-v)) * f0 Show: v/c =...
  10. M

    Help with Geometry Proof: Find CD in terms of AD and BD

    I need help in how to do this proof. A circle is given with diameter AB. pick any point C on the circle and drop a perpendicular from C to the given diameter at D. Find CD in terms of AD and BD.
  11. M

    Proving Limits of Function f(x) = x^3/abs(x)

    I have the function f(x)=x^3/abs(x) I think that the following are all true: lim f(x)= inf. x->inf lim f(x)= 0 x-> 0+ lim f(x)=0 x-> 0- lim f(x)= -inf. x-> -inf and lim f(x)= dne. x-> 0 I'm not sure about the last one, because I thought that ususally when the limit...
  12. M

    CD: What Makes a Spacetime Geodesically Complete?

    I'm reading an article (http://arxiv.org/abs/gr-qc/0403075) which proves that a certain spacetime is geodesically complete. It does this by proving that the first derivatives fo all coordinates have finite bounds. My question is why this is enough. Is it just a simple ODE result? We know...
  13. B

    Question on proofs for a CS related class

    Homework Statement As a background to this...I have no experience with proofs at all. I did not take a formal geometry class in high school (took a shortened summer course that gave a VERY brief overview of proofs) and have not gotten to discrete math in university, so I really do not know how...
  14. G

    How Do Distinct Rows in Matrices Influence Their Rank?

    1. Let A be an m × n real matrix of rank r whose m rows are all pairwise distinct. Let s ≤ m, and let B be an s × n matrix obtained by choosing s distinct rows of A. Prove that rank(B) ≥ r + s − m. Solution: Assume that s is the largest amount of distinct rows of A. r = n-dimNul A...
  15. R

    I understand deltas and epsilon proofs for the most part

    so 0 < l x-a l < delta and l f(x)-L l < epsilon What I don't understand is how come deltas and epsilons can't be greater than or equal to their respective differences?
  16. N

    Disc. math/logic: division & modulus proofs

    Homework Statement Show that if a, b, c, and d are integers such that a | c and b | d, then ab | cd. Let m be a positive integer. Show that a mod m = b mod m if a ≡ b(mod m) Homework Equations | means "divides," so a | b means "a divides b" or "b can be divided by a" mod gets the...
  17. A

    Prove Number Theory Proofs: Sum Irrational, (m+dk) mod d, x^2=x, n^2 mod 3

    1. For any positive integer n, if 7n+4 is even, then n is even. 2.Sum of any two positive irrational numbers is irrational. 3. If m, d, and k are nonnegative integers with d=/=0 then (m+dk) mod d = m mod 4. For all real x, if x^2=x and x=/=1 then x=0 5. If n is an integer not divisible by 3...
  18. J

    Proofs of the Existence of No Greatest Natural Number

    Earlier today, I was thinking about the statement that "there exists no greatest natural number" and immediately, two proofs sprang to my mind. Since my question depends on these, I'll write them out below . . . Proof 1: Let n \in \mathbb{N}. Clearly n+1 \in \mathbb{N} and n < n+1. Since...
  19. H

    Are Books Necessary to Understand Mathematical Proofs?

    I'm looking for book about making proof. Is this kind of book even required to understand proofs? Is there some special theory behind proofs, or books about proofs just provide examples, and are more like "math for dummies" ? I'm not sure if it's proper to use that kind of book, should i...
  20. C

    Series Convergence/Divergence Proofs

    \sum_{n=1}^{\infty} n \sin(\frac{1}{n}) I rewrote the sum as \sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\frac{1}{n}} Then I applied the Nth term test and used L'Hoptials rule so \lim_{n\to\infty} \frac{\cos(\frac{1}{n})\frac{-1}{n^2}}{\frac{-1}{n^2}} The \frac{-1}{n^2} cancel...
  21. W

    Proofs involving subsequences.

    Homework Statement Which of the following sequences have a convergent subsequence? Why? (a) (-2)n (b) (5+(-1)^n)/(2+3n) (c) 2(-1)n Homework Equations Cauchy Sequence Bolzano-Weirstrass Theorem, etc. The Attempt at a Solution (a) The sequence I get is...
  22. silvermane

    Combinatorial Proofs of a binomial identity

    Homework Statement Show that for all integers n,m where 0 ≤ m ≤ n The sum from k=m to n of {(nCk)*(kCm)} = (nCm)*2^(n-m) The Attempt at a Solution So for the proof, I have to use a real example, such as choosing committees, binary sequences, giving fruit to kids, etc. I have been...
  23. silvermane

    Combinatorial Proofs of Binomial Identities

    Homework Statement (Give a combinatorial proof of each of the following identities. In other words, describe a collection of combinatorial objects and then explain two different methods for counting those objects. Leave each identity in the form given. Do not rearrange terms or use any other...
  24. S

    Solving Hard Calculus Proofs with Steve

    I've got two calculus proofs that I can't seem to get! I was wondering if you guys could help me out a bit... 1. Homework Statement Suppose x_{n} is the sequence defined recursively by x_{1}=0 and x_{n+1}=\sqrt{5 + 2x_{n}} for n=0, 1, 2, 3, ... Prove that x_{n} converges and find...
  25. M

    Understanding Prime Power Proofs

    Hi, I am having trouble understanding this proof. Statement If pn is the nth prime number, then pn \leq 22n-1 Proof: Let us proceed by induction on n, the asserted inequality being clearly true when n=1. As the hypothesis of the induction, we assume n>1 and the result holds for all...
  26. Y

    Linear algebra proofs (linear equations/inverses)

    Homework Statement Two problems. (1) Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent. (2) Can you prove that the matrix A = [1 1/2 ... 1/n 1/2 1/3 ...1/(n+1) ... 1/n 1/(n+1)...1/(2n-1)]...
  27. C

    Physics Proofs: What to Expect as a Physics Major

    As a prospective physics major, I would like to know if physics is as "proof heavy" as math is, outside of the math prerequisites. Thanks!
  28. N

    Proofs of Coleman-Mandula & Haag-Lopuszanski-Sohnius Theorems

    Does anyone know of any sources which provide a proof, or outline of, the Coleman-Mandula theorem and the Haag-Lopuszanski-Sohnius Theorem?
  29. M

    Linear Transformation Proofs: Check My Work and Correct Errors | Math Help

    Hi, would someone be able to check my proofs for me and tell me if they are right and if not what is wrong please? So for the first one I said let u=p(x) and v=b(x) T(u+v)=p(x)+b(x)=p(5)x2+b(5)x2=Tu+Tv and T(ku)=(kp)(x)=kp(5)x2=kTu So it is a linear transformation. For the second I said...
  30. M

    I'm going to fail my first proofs class. How do people even learn this?

    I'm going to fail my first proofs class. How do people even learn this!? A bit of cheese with my whine perhaps but I'm more frustrated because I don't know what to do. I am going to fail this class unless I figure it out before our first exam. I read the chapters, take notes, try to understand...
  31. C

    Fibonacci Proofs via Induction

    So I am looking at the following two proofs via induction, but I have not a single idea where to start. The First is: 1. Suppose hat F1=1, F2=1, F3=2, F4=3, F5=5 where Fn is called a Fibonacci number and in general: Fn=Fn-1+Fn-2 for n>/= 3. Prove that: F1+F2+F3+...+Fn=(Fn+2)-1 Secondly...
  32. B

    Does Cross Cancellation Ensure a Group is Abelian?

    I have two proofs that I am uneasy about and one I'm having trouble with so hopefully I can figure out where I'm going wrong if I am. Ignore the weird numbers, its to help me organize the problems. 14) Let G be a group with the following property: Whenever a, b and c belong to G and ab=ca, then...
  33. C

    Proving with Delta Epsilon: A Beginner's Guide

    Hi there, I'm having trouble understanding how to prove things using the \delta \epsilon definition. I have read a few other threads and sites, but I can't seem to put it together. For example, I came across this problem, if given limx-->af(x) = L, how would I prove (using delta-epsilon and...
  34. N

    Formal Boolean Proof of A ⊕ B' ⊕ C = (A ⊕ B ⊕ C)

    Homework Statement Prove that A \oplus B' \oplus C = (A \oplus B \oplus C)'Homework Equations The Attempt at a Solution I tried to use A \oplus B' \oplus C = ABC' + A'B'C' + A'BC + AB'C But i am not sure how to proceed further from there... Please could someone give me a little bit of help ?
  35. H

    Understanding Proofs in Math: Tips & Book Suggestions

    I enjoy math, but when i have to proof something - i find it confucing. I don't know how strict proof should be. If proof ends with 17<18 , it this really end? Why shouldn't i proof that 17<18 is actually true? I never know what sentences can i treat as obvious, and which one i have to proof...
  36. J

    Proving Divisibility by Induction

    Here are some that I'm stuck on. Pg. 56, #12 Prove by induction on n that, for all positive integers n, 3 divides 4^n + 5 Of course, the base case it is P(1) = (4^1 + 5) / 3 = 9/3 = 3...TRUE! I just can't see the trick here. P(K+1)= (4^(K+1) + 5) / 3 = ((4)(4^K) + 5)/3= ... not...
  37. N

    I have no idea how to do (very very basic) proofs help guide me?

    Homework Statement Let A and B be any sets. 1: Prove A is the disjoint union of A\B and A intersect B. 2: Prove A U B is the disjoint union of A\B, A intersect B, and B\A. Homework Equations ? The Attempt at a Solution I understand most of the basic terminology used. I know disjoint...
  38. N

    Solving Isometries Proofs: Geometry Revisions & Help

    Have been revising geometry today and have came across some proofs that I can't seem to find in books, but I can't get through either. Any help would be great. Let A be a 3x3 orthogonal matrix and let x and y be vectors in R^3 a) Show that detA = +/- 1 b) Show that the length of Ax is...
  39. H

    Help I dont understand Geometry Proofs at all

    Homework Statement I am home schooled and I am having a really hard time with proofs! I keep going back over stuff and reviewing but every time I just feel more lost than before! Any advice or help?
  40. D

    Trouble with delta epsilon proofs

    i know how to do basic proofs, but some proofs on the actual limit theorems confuse me. my textbook's choices for delta are very obscure and i have no idea how they even came up with them. for the proof of the limit theorem where the limit of a product of 2 functions is equal to the product...
  41. S

    Laplace Transform Proofs: Get Help Now

    Hey, I have been studying differential equations a bit and was wanting some help on some proofs. There are 3 laplace transforms I would like proofs for. Not really sure where to get started or if someone could lead me to place that has these proofs I would greatly appreciate it. Thank you.
  42. R

    Linear Algebra Proofs for nxn Matrices | Homework Assistance

    Homework Statement Ok so I am stick on three proofs for my linear algebra final adn help on any of all of them would really help with my studying For the first 2 assume that A is an nxn matrix 1.If the collumns of A span Rn then the homogenous system Ax = 0 has only the trivial solution...
  43. D

    I'm terrible with proofs linear algebra problem

    let { v_1, v_2, v_3, ... v_m } (m > 3) be linearly independent vectors in a vector space V. Prove that the set { v_1 - 2v_2 + v_3, v_2, v_3, ..., v_m } is also Linearly. I did this: a_1( v_1 - 2v_2 + v_3) + a_2(v_2) + a_3(v_3) + ... + a_m(v_m) = a_1(v_1) - a_1(2v_2) + a_1(v_3)...
  44. D

    Proof of lim (x to 0) of sinx/x and circular proofs

    I wish to prove \lim_{x \to 0} \frac{\sin x}{x} = 1 using L'Hôpital's rule. The problem with this is, even though the result after applying the rule is 1 (the correct answer), the limit itself was assumed to be correct in order to calculate the derivative of sinx. This constitutes circular...
  45. S

    Proofs of subspaces in R^n (intersection, sums, etc.)

    Homework Statement Let E and F be two subspaces of R^n. Prove the following statements: (n means "intersection") If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors Note: Above zero denotes the...
  46. moe darklight

    Alternative Proofs To Euclid's Propositions

    So lately I've been trying to start practicing proving things without being given a prompt (as in actually finding something out, not answering the question: "prove so and so"), which I'd never done before. -- But then the bummer is not knowing the name of what you've just proved :rolleyes...
  47. O

    Proving Elliptic Orbit with Rotational Matrices

    Homework Statement Prove that: r=a(cos E-e)(ihat,xi)+(sqrt(a*p)) *sin E (ihat,eta) Homework Equations E=eccentric anomaly e=eccentricity The Attempt at a Solution Rotational matrices come into play here, but I'm not sure to what extent. alpha=beta*gamma*delta, with their...
  48. M

    Can Linear Algebra Prove Vector Dependencies and Transformations?

    1) prove that for any five vectors (x1, ..., x5) in R3 there exist real numbers (c1, ..., c5), not all zero, so that BOTH c1x1+c2x2+c3x3+c4x4+c5x5=0 AND c1+c2+c3+c4+c5=0 2)Let T:R5-->R5 be a linear transformation and x1, x2 & x3 be three non-zero vectors in R5 so that T(x1)=x1...
  49. F

    What is your thought process as you do proofs?

    Just wondering. I haven't been having problems with proofs, so far, but I'm interested in how people think about proofs. I feel I'm still far from ideal. There are some standard proofs, like when proving uniqueness which have all looked the same so far. There are also counting proofs, in which I...
  50. M

    What is the relationship between invertible linear mappings and rank in proofs?

    1. Hi! I was wondering if anyone could help me to solve the following problem! Let L : [R][n] ->[R][m] and M :[R][m]-> [R][m] be linear mappings. Prove that if M is invertible, then rank (M o L) = rank (L) thanks! :)
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