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 you effectively approach writing a proof?

    I find proofs very difficult. What process would you go through before writing a proof? What do you do generally?
  2. J

    Using four-vectors in derivations and proofs

    I'm taking a first course in modern physics and were currently discussing special relativity. My professor keeps using four-vectors in derivations and proofs, and requires us to use them, but he hasn't developed the theory behind them; that is he's only showed us how to manipulate them. The...
  3. S

    Trouble With Trig Proofs: Seeking Help

    I recently posted about some trig equations, now I'm doing some HW on trig proofs, i got the first couple trig proofs, but had trouble with the last two. Here are the two problems (attached). For the first one, i can't even get started. i have some ideas, but i can't find out how to get the...
  4. S

    Constructing Proofs: Solving Set Functions with Characteristic Functions

    Constructing Proofs help! Here is the problem: Given a set S and subset A, the characteristic function of A, denoted \chi_A, is the function defined from S to \mathbb{Z} with the property that for all u \ \epsilon \ S: \chi_A(u)= \begin{cases} 1 & \text{if u $ \epsilon \ A$} \\ 0...
  5. P

    A math book on introducing proofs(?)

    Hey guys… and girls! I was just wondering if anyone knew of any good books that introduce the concepts and reasoning behind mathematical proofs, starting from the beginners level. (In high school my teachers did not emphasize proofs.) I would like this specifically to help me for first year...
  6. M

    Good books for logic, proofs course?

    I have to teach the "bridge" course for junior level math and math ed majors on proofs and logic, and need to find a book. I do not like books that are mathematically vacuous. I.e. I want one that teaches how to prove things and then actually proves something of mathematical interest...
  7. M

    How can I prove [(A^B)-(B^C)]-(A^C)'=0 using contradiction?

    I'm trying to prove the following by contradiction: [(A^B)-(B^C)]-(A^C)'=0. A, B, C are sets. All I know is in order to prove by contradiction you simply set the above not equal to zero. But I don't know where to go from there. "^" means the intersection symbol.
  8. honestrosewater

    Proofs about weights of wffs

    Definitions: Briefly, for the formal, object language L, there are two mutually exclusive categories of primitive symbols: (i) an infinite set of propositional symbols and (ii) two distinct connectives, ~ (negation) and -> (implication). If s_1, s_2, ..., s_l are (not necessarily distinct)...
  9. RadiationX

    Schools Community college and the lack or teaching proofs

    community college and the lack of teaching proofs I'm in calculus II and to this date i have never had to write one proof! when i look through the forums i commonly come across postings about how to prove things, even from high school kids. why are community colleges less rigorous than 4 year...
  10. N

    Need some help with basic complex analysis (no proofs)

    need some urgent help with basic complex analysis (no proofs) This forum is probably more appropriate. please forgive me for double posting. Can someone give me examples of the following? (no proofs needed) (C is the complex set) 1. a non-zero complex number z such that Arg(z^2) is NOT...
  11. N

    Need some help with basic complex variables (no proofs)

    need some urgent help with basic complex variables (no proofs) Hi: can someone give me examples of the following? (no proofs needed) 1. a non-zero complex number z such that Arg(z^2) "not equal to" 2 Arg z 2. a region in C which is not a domain 3. a non-empty subset of C which has no...
  12. B

    Mathematica Which Book on Mathematical Proofs is Perfect for Preparing for Grad School?

    I plan on going onto grad school at some point in the near future and I know I could use a ton of work in the area of constructing proofs. What I'm looking for is a book that could shed some light on how this process is approached. That is to say for example maybe how a mathematician would...
  13. N

    How to Prove Divisibility in Math Problems?

    i just started my second semester with geomtry and am having difficulties with these proofs. i am stuck on this one question which asks: prove that if n is an odd positive integer, then one of the numbers n+5 or n+7 is dividsible by 4. so this is what i came up with: let n = 2k+1...
  14. F

    Solving the Multivariable Proof: A+C/B+D < E+G/F+H

    I'm confronted with the following question that may of may not have a solution: You are given eight variables, A, B, C, D, E, F, G, and H. These variables are integers. You know that: A/B > E/F and C/D > G/H Is it possible that (A+C)/(B+D) < (E+G)/(F+H)? I've tried...
  15. C

    Indirect Proofs: Shaping the Proof

    Hey, anyone ever done indirect proofs? Maybe my school is a little weird, but we are doing those. IF you did, how do we shape the proof?
  16. L

    Mathematica Books on Mathematical Proofs and Theory

    I'm looking for a book that gives you many equations and goes through proofs etc. One of my weaknesses mathematically tends to be logically getting from one point to another when I'm not solving problems numerically and remembering what are and what are not legal steps to prove something. I'm...
  17. M

    Proofs in sequences and series

    I am teaching honors calculus in college, and trying to teach something about convergence of sequences and series. my class has apparently never seen a genuine proof in high school and have no idea how to begin one (answer: with the definition). I have had students ask me what "QED" stands...
  18. C

    Fundamental mathematic proofs

    Fundamental mathematic proofs... I know this may seem a slightly odd question, but are there any website or pdf files, etc, floating around of proofs of the basic pricipals and "tricks" of maths? eg - adding, subtraction, multiplication, division, fractional sums and products, percentages, etc...
  19. T

    Proofs for -1 = 1: Exploring the Problem

    heres a little problem that at a first glance is real: \frac{1}{-1} = \frac{-1}{1} so \sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}} by splitting it the square root into two parts... \frac{i}{1} = \frac{1}{i} and i^2 = 1 -1 = 1 wonder if there are any more similar "proofs"?
  20. I

    Tutoring a Math Student: Struggling to Write Proofs

    I'm tutoring a girl in my math class on how to write proofs. She understands what information she needs to prove something, but the only problem is she doesn't understand how to put the data in order. I tried to the following to clear things up for her: 1.) I asked her to prove...
  21. K

    Tips for doing proofs in calculus

    Hi, I have a major test next week and some questions will be on epsilon and delta proofs. From the homework I have done, these epsilon and delta proofs can be applied anywhere and in any scenario. Therefore, I was wondering do you guys have any tips on handling and solving these questions...
  22. D

    Grade 12 Physics: Practice Equation Proofs for Work, KE, Springs, & Potential Energy

    I am in grade 12 physics, and i have to practice equation proofs. I am currently studying work, kinetic energy, springs, and potential energy (gravity and elastic). Does anyone have a good proof?
  23. M

    Proofs of a God or no God is pretty much useless?

    Proofs of a God or no God are pretty much useless? I sometimes find myself staring at the absurdity of looking for a proof for the existence of a God, or the proof for the non-existence of one. My logic is pretty simple, say if we say that a orderly universe implies existence of a God, but...
  24. M

    Mastering Proofs: Tips & Examples

    I was just wondering, since i m kind of weak in doing proofs, what is the best way of understanding on how to do proofs. What is the best way to master, if one can, on doing proofs? or even if not master, but to be able to do proofs without "thinking", like sometimes my teacher says he just does...
  25. D

    Recursive sequence problem: proofs by mathematical induction

    Guys, I'm trying to prove by induction that the sequence given by a_{n+1}=3-\frac{1}{a_n} \qquad a_1=1 is increasing and a_n < 3 \qquad \forall n . Is the following correct? Thank you. :smile: Task #1. n = 1 \Longrightarrow a_2=2>a_1 is true. We assume n = k is true. Then...
  26. E

    Proofs involving prime numbers

    We didn't talk about prime numbers in my class, but several of the homework problems mention them. For instance: Prove that if every even natural number greater than 2 is the sum of two primes, then every odd natural number greater than 5 is the sume of three primes. Assume that n is an...
  27. E

    Need help with two simple proofs

    Here's my problem: Provide either a proof or a counterexample for each of these statements. a) For all real numbers x and y, if x is greater than 1 and y is greater than zero, then y^x is greater than x. My proof: Suppose x is some real number greater than 1 and y is some real number...
  28. A

    Problems involving finding proofs?

    Where can I find some good problems involving finding proofs? I want to see how far I can go as it's relatively new to me...
  29. M

    Proofs advice/places to find more practice proofs

    Hey I have 2 quick questions... 1) Any advice for proofs, I am just starting with them, and wondering how I can make good proofs, i know very little about the now :| anything that other people expierienced while starting proofs would be great, (its grade 12 algebra) 2) Anyone know where...
  30. E

    Set Theory Proofs: f:X->Y Function and Subset B of Y

    Let f:X->Y be a function 1) Given any subset B of Y, prove that f(f^-1(B)) is a subset of B 2) Prove that f(f^-1(B))=B for all subsets B of Y if and only if f is surjective Help anybody?
  31. F

    How to Learn Math and Do Proofs

    Hello everyone, I'm a budding theoretical physicist and mathematician, all throughout my education I've been taught about mathematical objects, relation between objects, Proofs, etc. Never have I been taught HOW to actually learn math. I've put together a website with all the lessons on how...
  32. C

    How does Archimedes' Principle prove buoyancy?

    Ok, I've seen many proofs of this, all being the same, but the closest I could find online was here: http://freespace.virgin.net/mark.davidson3/IMS2121/buoyancy/Buoyancy.html Basically the idea is you mess around with the formulas for pressure and hey bingo. However, I have one question - the...
  33. R

    What are epsilon-delta proofs?

    Thanks in advance for any help, I'm trying to understand epsilon-delta proofs, and the various sites I've found so far aren't helping that well. I know that epsilon is referring to a small number >0, and delta traditionally refers to a number > epsilon, but I'm not quite sure of why this...
  34. E

    Proof of Limit: |Rez - Rez0|<E Whenever 0<|z-z0|<D

    How do I show |Rez - Rez0|<E whenever 0<|z-z0|<D is true, where E and D are real number greater than 0, and z is obviously a complex number? In other words, proving that the lim of Rez (as z approaches z0)=Rez0.
  35. L

    Proofs on the rationality of pi

    Im looking for some proofs on the rationality of pi. I also want to know what some people think about it.
  36. Q

    Enjoy!Can the Sum of Reciprocal Powers be Represented by an Infinite Product?

    Know any 'nice' proofs in maths? Or know an alternative and simpler/nicer proof to common method employed? Post here ==>
  37. M

    Is 0.999... Really Equal to 1? Exploring the Mathematical Proofs

    I'm sure most of you already know this. The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1. Here are the...
  38. R

    Mathematica Do mathematical proofs exist, of things that we are not sure exist?

    Do mathematical proofs exist, of things that we are not sure exist, especially those, that do not have observational confirmed data?
  39. B

    Proving Trigonometric Identities: Solving Challenging Pre-Calculus Problems

    I know this is below most of those that peruse these forums, but I've been giving myself an ulcer trying to figure these silly things out. The first problem starts out as (1-sin^2(x))(1+tan^2(x))=1 and I've got it down to (sin^2(x)/tan^2(x))-sin^2(x)=1 but from there I've got no idea...
  40. M

    Proofs for Superstring Theory: Tests & Variance

    Ok, I know about the search for supersymmetric partner particles (sparticles) and the tests on gravity variance at small scales, but what other tests are there that can be used to add proof to superstring theory?
  41. M

    Looking for a few good proofs

    I understand how to use such things as product rules, quotient rules, parts by integration, but it bothers me I don't really have a deeper understanding of it. My book offers rather rigorous proofs, they are all pretty much: assume this to be this and let this be that so it must equal this...
  42. R

    Proving Parallelogram PQRS & Quadrilateral ABCD: Help Needed

    I'm having trouble with two parallelogram proofs 1) PQRS is a parallelogram and T is any point inside the parallelogram. Prove that triangle TSR + triangle TQP = 1/2 parallelogram PQRS 2) ABCD is a quadrilateral whose area is bisected by the diagonal AC. Prove that BD is bisected by AC...
  43. W

    Proofs: If a|b then -a|b, a|-b, -a|-ab

    Hello, First I will post the question that I am working on. I am not good at proofs (even elementry proofs such as these ones). I was wondering if someone could take a look at my work and perhaps confirm whether my proofs are adequate and/or make some suggestions. First I will start...
  44. M

    Vector Proofs in geometry

    I was having some serius problems when proving some of the questions where we are given, let's say, a rectangle, there is one diagnol, and the other diagonal is connected to a line that is in a ratio, and the diagnal connects to the point that divides that line. The concept is combined with...
  45. W

    Can I Use Theorems In Geometry Proofs?

    [SOLVED] Geometry Proofs. Hello, I am currently taking a second year mathematics course in geometry at university. I have to do quite a few proofs and I am not used to doing proofs much less geometry proofs -last time I took geometry was when I was in grade 10 and that was over ten years...
  46. L

    Vector Proofs using vector components

    Hi! I'm new to the forums. I'm taking an introduction to physics class this semester and I've been having some difficulty with it. Oh, I also wanted to let you know that it's been a while since I've taken calculus or any other math class for that matter. But I need physics to graduate. Anywho...
  47. H

    Proving Prime Numbers: Understanding the Non-Divisibility Theorem in Mathematics

    Hello everyone, My first post on these forums and I was wondering if I could have some assistance/direction with a problem: Prove that if p is a prime number and a and b are any positive integers strictly less than p then a x b is not divisible by p. The first thing I thought to myself...
  48. K

    Back Induction: Proofs Beyond AM>=GM

    Backward induction Is there any proof that involve the use of back induction besides the proof of AM>=GM ? It is the only example I've come across that use back induction.
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