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Usign postulates to prove validity of a theorem

  1. Aug 30, 2006 #1
    Hello,

    recently I got this question on my Logic Design course exam,

    Prove the following theorem using postulates:

    X OR 1 = 1

    I explained it using a simple argument,

    a variable can be considered as a switch in a circuit, if it is 1 then the switch is on, if it is 0 then the switch is off. The OR operation acts like if we would have switches wired in pararell, since in the case of (X OR 1) one switch would be always on, thus the current would always flow which means that the output of X OR 1 would always be 1 no matter of X, thus we have shown that the statement is obviously correct.

    Strangelly the person grading the exam wasn't very pleased with this explanation, guess it was to "obvious", I got 0 points whitch I don't like. Anyway, I am asking

    are you allowed to prove theorems with postulates?, Since a postulate is NOT an axiom, that means that the postulate is not necessary something that is a perfect choice to do profs with, axiom is. I understand why would I want to prove things with axioms, but why would I want to prove things with postulates??? Doesn't make sense to me.

    Just one more thing, I noticed that on the exam, the gradient symbol(upside down triangle) was used to denote XOR operation, is that allowed? Isn't gradient a reserved symbol and should not be reused for something other than gradient?
     
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  3. Aug 30, 2006 #2

    matt grime

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    So you didn't prove it as you were required to do, hence you go no marks. It is not strange at all.

    Axioms and postulates are, mathematically, synonymous terms in my opinion.

    You can use any symbol for whatever purpose you want. There are no reserved symbols (maths is not computer science, nor is it electrical engineering for that matter). Of course you'd be shooting yourself in the foot for making bad choices of symbols. However, there is zero chance of being asked a question about grad in a logic course so there is no possible confusion at all that can arise.
     
    Last edited: Aug 30, 2006
  4. Aug 30, 2006 #3
    Thanks for the reply.

    It is a shame that I have a mathematician for my Logic Design course. Soo the purpose is not understanding but knowing the Math dogma and doing things the mathematicians way. Guess that weather or not I can imagine and understand the theorem is not important but weather or not I can memorize some postulates and do profs with it with the least amount of thinking. Well done!

    Btw: Why would you want to prove something with a postulate that it(the postulate) cannot be proven? Proving things with things that cannot be proved is not a good argument in my book.
     
  5. Aug 30, 2006 #4

    matt grime

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    Oh, dear. Look, what is a proof? Why do you know that every time you flick that switch the electrons will do what you presume they will? Why is OR 'like a switch in parallel' have you proved that 'variables are switches in circuits'?

    A proof is a proof is a proof. A heuristic argument is an analogy and demonstrates nothing other than that something is plausibly true based upon your experiences and belief in the analogy being true.

    The purpose of the question (or more accurately your answer) is to demonstrate that you know what the axioms of logic are and can make deductions from them in a logical way. I.e. that you understand what logic is, not that you understand something about electrical engineering.

    You also have a misapprehension of what 'axioms' actually are. Search these forums for plenty of discussion about why it is wrong to state an axiom is a self evidently true statement. If you don't want to learn about proof and logic and maths fine, but don't moan about the subject.
     
    Last edited: Aug 30, 2006
  6. Aug 30, 2006 #5
    Guess what you want to do is to prove something obvious in the most unobvious way. Why would you want to do that? My point was that it is very obvious that X OR 1 will always result in true. That can be easy shown with the circuit analogy. Soo I didn't prove it the Mathematical way, but just shown that it is darn obvious that it is true in any case. If you can't use your common sense when doing Math that is a shame, don't you think?

    I could have proven that usign YOUR way but don't you think it is a waste of time proving things that are obvious?

    I would understand if I was asked to prove the consensus theorem since it isn't very clear and obvious that it is true in any case. I would enjoy proving that.

    Axioms as I see them are nothing but DOGMA!!! You have a statement and say this is an Axiom as such it must be TRUE, soo you assume it is true and take that for granted. What I hate is that you are not allowed to question the validity of axioms. But questioning the validity of axiom wouldnt make any sense, would it?

    I lose all my interest at the exams when I am asked to prove something that is soo obvious... Damn it, I am an Engineer not some Mathematician, if you have fun at proving the most obvious things, please do soo, but please don't bother me with that stuff.

    BTW: Why doesn't the physicist then bother people with proving the validity of the 2nd Newton Law? The laws of electromagnetism? The laws of thermodynamics? What postulates would you use to prove that? Why is that soo?
     
    Last edited: Aug 30, 2006
  7. Aug 30, 2006 #6

    HallsofIvy

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    It is not that you must use a specific kind of proof but rather that an analogy is not any kind of proof. Your original "proof"
    just tells about how "XOR" switches are set up, saying nothing about WHY those switches give "XOR" which was the whole point of the question.

    No, it's a good way to start learning HOW to prove things as well as a good why learn WHY it is obvious. You seem to be saying you want to learn HOW to do things without learning WHY that is the correct way to do them. In any case, if you think something is "obvious" but can't think of a logical way of proving it obvious, all you are saying is that you have always assumed it was true. "Obvious" truths is just another phrase for "prejudices".

    But since you have asserted that you don't know what a proof is, I doubt that you could do it.

     
    Last edited: Aug 30, 2006
  8. Aug 30, 2006 #7

    matt grime

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    well, I certainly agree it is a trivial result, but the question didn't ask you to draw any analogies, nor did it claim to be a hard result. It is checking that you understand the material in a simple example and can present an cogent mathematical argument in a very simple case. I would argue that the question is too simple and leads exactly to the problems that people can't see that there is anything to prove at all.

    you have failed to understand why you were being asked the question.

    As I said, you have a very big misconception about what axioms are. Axioms are *not* unquestionably true. They are simply a set of starting propositions in a system from which other results can be deduced.

    CANONICAL EXAMPLE that axioms are not 'absolutely true': the parallel postulate in geometry states that given a line and a point not on the line there is a unique parallel line through the point. In Euclidean geometry we take it as an axiom, in spherical and hyperbolic geometry we do not.

    You are prefectly at liberty to question, alter, replace, negate or do anything you wish to axioms. What will result is another theory. Now, since you're so keen on your circuits, what happens when you change the axioms of logic you're dealiing with means that the circuits you're happy to play around with cease to be a model for that system of logic.


    no one would dispute the question is easy.

    I assure you mathematicians would find that question even more insulting than you do. But the point is to test that you can write a formal proof and understand logic.

    You can't prove a phyiscal theory. You can only experimentally validate it or disprove it.
     
    Last edited: Aug 30, 2006
  9. Aug 30, 2006 #8
    I didn't claim that I have proved the thing with the analogy!!! All I claim is that it is obvious that the thing is true which can be shown with the analogy, nor would I ever consider doing proofs with analogys.

    Btw: XOR is the exclusive OR operation, I was refering to OR operation.

    I just don't find any challege in proving trivial things.

    Thank you for the vote of confindence.

    I didn't get that feeling, but thanks for leting me know.


    Please, just because I don't find challenge in proving the most trivial theorems that doesn't make a mediocre engineer. And my engineering skills have little to do with proving mathematical theorems.

    You are wrong. If you prove that the 2nd newton law is correct in an experiment that included a 2kg object, then the 2nd newton law is only confirmed to be valid for a 2kg object under that circumstances but it is not generally proven. Matt you would probably back me up at this one? Since in the "lab" you have only proven that e.g. 2nd newton law is correct under the lab circumstances, what gives you the right to assume that you would get the same results if you did the same experiment onboard the International Space Station? All what you do in the lab is to see if the physical theory confirms to nature. As far as energy goes, it is nothing but a mathematical concept. Read the chapter about energy from the Feynmans lessons on physics, it is superbly explained. I do assure you we Computer "Engineers" do have phyisics lessons.
     
  10. Aug 30, 2006 #9
    Thank you very much for the reply Matt,

    Precisely, well I guess next time I would have to turn off my ego for one moment, brace myself, and do the proof. But nevertheless I will still consider it a waste of my time.

    Yes! I don't like Math questions in an engineering exam, I have too many of those from the regular math lessons.

    Thanks for the explanation. I would just like to see that the focus of Math and engineering lessons would be more on USING the theory rather than PROVING the theory. Since I get the felling that for a Math person proving something is more important than using something. I find it more rewarding that I am able to e.g. calculate what would be the temperature of a body that is cooling under certain conditions after 6 hrs, rather than being able to prove something since proving something in my book is doing nothing of a consequence. What good is to prove something that was already proven zillion of times before? That doesnt mean that I am not capable of proving something, I just don't see the point in proving something that was already proven zillion of times and it is darn obvios. It is more rewarding for me to be able to calculate something and that I can actually see the result of the calculation in the physical nature. It is rewarding for me to calculate a somewhat complicated equation involving some flip-flops and then realizing that circuit and seeing that the physical nature - the circuit, confirms to the behavior described in the equation. But doing proofs is just a wast of time for me. It saddens me that I am forced to waste my time on that.

    Ok, guess I have no chance on passing the lessons without subduing to the will of the mathematicians, I will try to brace my ego and do the proof next time.

    Ofcorse, porhaps this is the part of my problem, I consider Math to be physical theory. That sounds strange.
     
  11. Aug 30, 2006 #10

    matt grime

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    You think doing these maths questions is insulting your ego is somewhat beneath you. The odd thing is that mathematicians like me would find the notion that you would like to simply put some numbers into the black body radiation formula hard to understand. That simple, mindless, pointless calculation you want to do we would consider beneath us. Anyway, all the calculations yo'ure doing have been done 'zillions of times before' as well. Just putting different numbers in does not make it a different question, just a different example of the same question.

    Anyway, don't forget, that you need to walk before you run. This is, I imagine, your first course in logic. Thus this question is somewhat akin to spelling 'cat'. Why turn off your ego? It is is a trivial question. Just answer it. Just think of it as the mathematical equivalent of 'do you use Fleming's left hand or right hand rule for induced current?' (Right, by the way.)

    You also shouldn't walk away thinking that it is in anyway reflective of what mathematics is. However, the point of an exam is to get some numbers that allegedly correlate to understanding of the material. It is not supposed to be an interesting walk through the mathematical scenery.
     
  12. Aug 30, 2006 #11
    Hope you didn't missunderstood me, with my ego I meant my stubborness.

    Well I find it amazing to see that you can put values in an equation and from it you can actually get the results that confirm to the physical observation. You find that beneath you? I find it amazing to be able to calculate even trivial things like how long would it take for a ball with given charasteristics to fall from 30 meter of height (including the air resistance in the calculation), for you this might be just another number pointlessly calculated, but I find it amazing that I could actually go to a 30 meter high building and drop the ball and observe how long it takes the ball to actually hit the ground. And then see that the value given by the equation and the one given by the observation differ by just a small amounth, I find that just amazing. But for you that is just some pointless equation given by some pointless person, for some pointless, useless calculation that was done zillion of times before??? I don't see it that way.

    It is funny that a lot of things in Math were invented by Physicist for the purpose of modeling the real world. Like Calculus, it wasn't invented soo you can play with some theorems, it was invented soo you can accurately calulate the volume of a barrel and more advanced things.

    What is the purpose of spelling the 'cat' rather than just the spelling itself? I see that as pointless. Why don't you see that as pointless task? What is your motivation for this? Do you find it amazing to be able to prove a trivial theorem?

    Why do you mathematicians enjoy proving the theorems soo much?

    I have explained why I like calculations soo much, but not just calculations, I find it amazing for example that from the definition of work and the 2nd newton law you can derive the equation for the kinetic energy. But this is just a trivial thing but I like is soo much, because there is something behind it. You can see that the equations have some physical meaning(consequence). Also amazing that you can get the mass of the Earth just by some trivial algebraic manipulation of the law of gravitation. In short there is something behind it!

    But with proving the theorems there is NOTHING to see!!!!

    That is just a shame. I hate it. I actually like mathematics but I don't enjoy one minute of the math lessons. They are just plain dry, the exam is even more dry and unimaginative as the lessons.
     
  13. Aug 30, 2006 #12

    matt grime

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    You are doing dull mathematics. That is all. You are drawing conclusions about the whole of mathematics, and of mathematicians, from your very limited and prejudicial experience. As I have repeatedly said the question you were asked to do is trivial, it is contemptibly easy, but people do struggle to be able to write mathematics properly so you have to start the learning somewhere, and if you can't do it then you've not learnt the material required of you. So shut up with the whinging and just do it. That question has bugger all to do with what mathematics really is.

    You're welcome to your opinion of what mathematics is, but it is based upon very little knowledge and is completely ill-informed. You like putting numbers into a formula that someone else derived and that you might well not understand. Fine. Hardly intellectually demanding is it? I didn't say the derivation of the equations was pointless, nor the equations themselves, but there is no stimulation to be gained from simply plugging some values in, is there? (Yes, I am taking this to an extreme and playing devil's advocate, but your attitude has driven me to this in an attempt to show you that there are different views on the world of mathematics and engineering than yours.)
     
    Last edited: Aug 30, 2006
  14. Aug 30, 2006 #13
    There are plenty of false statements that I can "prove" via analogy. I’ve seem some very convincing analogy based “proofs” that pi is the root of an algebraic equation or that 1=2.
     
  15. Aug 30, 2006 #14
    Well unlike in mathematics you cannot prove anything in physics, at most you can demonstrate that the results are in confirmation with experiments. Any physical theory that cannot be falsified because "it is proven to be correct" is highly suspect IMHO.
     
  16. Aug 30, 2006 #15
    I wasn't pointing at Mathematics as the subject, but rather at the mathematical pedagogy in general. Asking somebody to prove something using the postulates of logic is cleary a 100 % mathematical question and no question about that. Why do you find it necessary to bug people with this formalism. Math as presented by Mathematicians in general (High School, College) is just a load of formalism and indoctrination and I don't like it at all and probably there is a load of people who don't particularly enjoy the Math lessons aswell. You say that proving something is Stimulating? Yet you probably learnt how to do proofs on some examples and then you use the same technique to prove other things! Like Mathematical Induction - it is a rutine, no stimulation, it is just like putting numbers into an equation, but I do admit, it is not that transparent as plugging numbers into an eqution but it is close enough not very intellectually demanding is it?

    From my limited experience I would conclude that Mathematicians are nothing but self absorbed egoistic hippocrits who think that they have all the answers in the world with their perfect formal axioms and theorems and that they can prove everything.

    Sorry about that but I have really bad experience with my High School and College professors, they all explained things in Math in the most complicated and formal way possible. And there were incidents like I got all the correct answers to a problem yet 0 points just beacuse I derived my own way of solving problem(I admit it was rather ad hoc but it did got me the correct solution!) rather than blindly following the technique explained by the Math teacher, then it is stuff like I write a decimal number as the answer, NO that is not correct 2*sqrt(3) the correct answer, Arrgh, and there was one task that had to do with cooling, soo I converted from minutes to hours and the solution I have put it to 2 decimal places precision since if the input data is of 2 decimal places precision then the output data cannot be of higher precision, that didn't go very well. Anyway, I will be a very happy man when I will have no more Math or Math like lessons. I know I am pathetic, but hey I am really having bad experience with Math people trought the course of my education probably I am not the only one.
     
  17. Aug 30, 2006 #16
    Precisely! That would be my view aswell.
     
  18. Aug 30, 2006 #17
    Matt from what you wrote I would assume that it is your opinion that doing proofs in math is stimulating in contrast to the pointless calculations? Not to mention that mathematical proofs show the proficiency of someone with the Math formalites and stuff.

    But if you take a look at lets say a Logic textbook, you will notice that there are a couple of done proofs, examples if you will, on how do you do it. It exaplains the exact procedure and notation on how you prove a logical theorem. Soo all what one is required to do in order to prove another logical theorem is to memorise the exact same procedure and just apply it. It is a pattern if you will, you learn a pattern and apply it. Hardly intellectually demanding is it? Same goes for other proofs, you learn a technicque(like Mathematical Induction) on how to prove something from a couple of examples and all what you do is apply the same technique to other examples. I don't find that more stimulating than doing calculations. Actually I find that a lot less stimulating that doing calculations, since the calculations have some meaning for me and proofs have none. There you go.

    I find mathematics nothing more than a study of patterns. And you make some techiques from that. Like the integration techniques. You don't have to understand anything! You just apply the same techniques as are explained on a couple examples in the textbook! Hardly intellectually demanding is it?

    Soo in general all what you have to do is learn some techniques and apply them blindy on the examples that are similar to the examples in the textbook. Soo you KNOW how to do it, but there is no understanding behind it. I would rather understant what one theorem means and how can that theorem be applied rather than knowing some proof technique and applying it blindly to the theorem.

    I think that Mathematicians value knowing more than understaning, which is bad. As I recall a colleague of mine who had quite nice grade in the Math coruse in one conversation he told me that he had no idea of what a derivate was, or what the second derivate was, he just tough of it as a magic number(pointless numbers as you put it) and he then just applied some techniques to it as the mathematician wanted and everything went fine. With proofs it was the same, you just apply the same technique as instructed by the Mathematician. You call that stimulating? I call that making people think they know a lot, yet they understand nothing!
     
    Last edited: Aug 30, 2006
  19. Aug 30, 2006 #18

    Hurkyl

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    You find it exciting to perform a calculation.

    I find it exciting to discover the calculation, and prove that it will work.
     
  20. Aug 30, 2006 #19

    matt grime

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    In that case you're drawing a concluicsion about mathematical pedagogy from one example, and a bad one at that.


    Who are we bugging? Do we do mass mail shots, email spam, put adverts in the newspapers? If you're wondering why your engineering based program chooses to make you learn maths 'properly' why don't you go ask them?


    Induction is one technique for proving a proposition. What is your issue with it? You appear to be confusing the proof with the thing it proves. And again you're extrapolating from your very limited experience of what you have been told is mathematics.

    There is no such simple definition as 'mathematics is...'. We don't have all the answers about mathematics, never mind anything else. I'm sorry you don't appreciate the culture of mathematics, but then that isn't my fault, or anyone else's.

    There, so far, appears only one egotist in this discussion, and that is you. (It would really also help you out if learned to spell correctly - if you're going to level charges at other people about them thinking they're perfect and egotists, I mean.)

    You do not have to blindly follow example. You just have to do things rigorously, and given your previous examples in this thread you do not do that so you get no marks. Did you prove your method was sound?Getting the correct answer for one question is neither here nor there. That perhaps is one lesson you can learn: in mathematics it is the journey that is important, not the destination. Further, if you refuse to speak French in a French oral exam you will get no points, so why should this be any different?


    And? You cannot write 2sqrt(3) as a decimal in a finite amount of time so of course you got marks knocked off: you wrote the wrong thing.

    No argument there, if you're relating the experience correctly, but then I did do physics.


    So, what's your point? That you dislike the culture of mathematics because you think it is attempting to state things as a physicist would, I surmise. Mathematics is mathematics. It makes no claims about what its conclusions say about the 'real world'. (Try dropping a feather from 30 metres and seeing if your equations hold, by the way.) In short, you are judging it by extrinsic values that it does not ascribe to.

    When Einstein wanted a theory of curved space he looked around and found out that one of those pesky mathematicians had written down the theory 30 years before there was any physical need to do so. That is one of the moral lessons the more applied people (well, you) could learn. Physics inspired the majority of modern mathematics starting in around 1800, but since then it is a subject in its own right, and is now beginning to feedback into physics. Number theory, of course, predates almost any physics there is, as does geometry.

    To draw a spurious analogy you 'hate' fiction because it is not non-fiction.

    All of modern computing is based upon propositional logic. Learn it, don't learn it, it's no skin off my nose, but drop the victim mentality about mathematics and look at the things that are out there. Start with trivial conjectures in number theory (in the sense of easy to state), and then try to find about those which are provable and those which are not. Why is there no nowhere vanishing smooth vector field on the sphere? What is the four colour theorem, and what is its proof?
     
  21. Aug 30, 2006 #20

    matt grime

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    No, that is not at all a defensible position.


    then your friend is a bad mathematician. Mathematics is not the blind application of formalism and rules. Do you understand why the axioms of logic you needed to know were chosen?

    No, you don't 'just apply the same technique'. To begin with, given a mathematical problem, there is no such thing as 'the techinique which must prove it'. You have to work out what you need to do to prove it. Which technique if any applies. Perhaps you need to invent a new one. Perhaps the proof will shed light on some apparently unrelated area. Perhaps a proof will need some idea from an apparently unrelated area.

    Of course merely to pass a course one only needs to memorize some things, but that statement is independent of the course: it applies as much to physics and engineering as mathematics, as well as chemistry. You don't need to understand the haber process to know that it's a good thing to mention if amonia is discussed.
     
    Last edited: Aug 30, 2006
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