what is principal of strong mathematical induction? in which situation this principle is used?
367692
If p(1) is true and, for each integer k>1 the statements  p(1),p(2),....p(k) imply the statement p(k+1) then p(n) is true for all.\n\nIn other words if a statement is true for the integers 1..k\nand you can prove it from those for k+1, than it is true for all integers.\n\nIt is a stronger version of normal mathematical induction where one use P(k) to prove P(k+1).
478213
Properties of mathematics makes it work. I'm not sure exactly what it has you do but essentially it does this.\n\nIf your number is X. You add 5 to it then subtract 5 from it and you get your original number. Of course the nice ones hide the fact that your doing exactly this. \n\nDavid Copperfield did a TV bit once where he had a 3X3 square. You start out at any square you want and move according to his directions. After each move he took away a square. In the end you always end up on one square. That was cool. Anyone heard of it?
312542
Ancient. This is the "Law of Superposition. See\n \nhttp://en.wikipedia.org/wiki/Law_of_superposition
144487
the answer is exponents.
924570
an exception.some thing not consistent with an assumed  theory/relationship-in maths, a point of failure of the equation
807703
Achieved Theoritical Upper Limit
460547
This looks close enough to the techniques used to show that the roots are close to the critical line. The actual estimates are difficult, though, and nobody has been able to actually give a proof using these techniques (obviously). Go ahead and try it. Perhaps you can see through to a proof. If so, you will be famous. If not, you will have learned somthing.
1163239
Have you heard of Occam's razor?\nIt says that to explain a framework, entities should not be multiplied beyond necessity.\n\nFurthermore, when multiple competing theories have equal predictive powers, the principle recommends selecting those that introduce the fewest assumptions and postulate the fewest hypothetical entities.\n\nNow, let's say God created heaven and earth. We introduce another element into the framework - God, that brings with it a whole new range of questions like Who created God, when did he start existing, what before him.\n\nThis doesn't simplify the argument, but makes it more complex. So, explain it with the least set of assumptions, being everything based on scientific principles which are all explained by mathematics.\n\nHope this is a logical and respectful argument.
922060
My suspicion upon seeing this question is that it can be solved using mathematical induction.  After all, couldn't we simply 'build' such a (maximally connected) graph starting with n=2 vertices and 'grow' it from there?\n\nRather than take on the proof myself -- mostly because I am unsure of the definition of 'components' -- I leave it as an exercise for you.  However, it bears noting that the formula shown is the commonly accepted "sum of the first n integers" formula used in almost every text on induction.\n\nGood luck!
806805
You can prove it by induction.\n\nIt is obviously true when n = 1.\n\nInduction pypothesis:  Suppose it is true for n\n\nWe need to show  is true for (n+1).\n\nDoes 24 divide (5^(2*(n+1))-1))?\n\nWe can rewrrite this as:\n\n25*(5^2n)-1  Subtract 24 from this to get 25*(5^2n)-25 \n\nThis is just equal to 25*(5^2n)-1).\n\nThis has to be divisible by 24 -- since one of the factors is (5^2n)-1) and our induction hypothesis says that it is.  \n\nAdd 24 to it and we get something that is also divisible by 24 -- but it is (5^(2*(n+1))-1)) -- so our induction is done.
974275
I had a professor who was always very adamant that you could not prove the hypothesis. Something about you don't actually test the hypothesis, you test the question....or somehting. He was very odd. I think there's a difference between a hypothesis and fact.
488404
If we imagine a line continuum between machine and man, and place Induction, Deduction & Analogy on it. Induction will probably be closer to the machine and Analogy will be closer to man, with Deduction probably in the center.
118732
good luck man... http://mathforum.org/library/drmath/sets/high_calculus.html
7873
well in the demostrantion of the Real numbers...\n\nall the numbers and relations depend of the existence of 1... suck that there are some rules..\n\nso we even demostrate that 1 exists... and that 1 > 0... after that we can prove the existence of any other Real number..
1179245
This is an inevitable occurrence; there is actually a theorem by Paul Erdos (I forget who he was working with) that is applicable...  Erdos-Schekeres Theorem, perhaps.  It's the one where they proved that if you take a piece of paper, cover the bottom of a box with it, remove it, wad it up and throw it back into the box, at least one point will be exactly above the same place it was before...  It doesn't take a whole lot of imagination to see a definite isomorphism between the E-S theorem (or whichever one it is) and what you're talking about.\n\nANYHOW, what it boils down to, in your case, is that you will have an experience (whether in dream or in reality is not germain) and it will match a future one.  On the whole, you should expect that to happen once in a while--and if you have a wide enough breadth of experience, perhaps more often...\n\nI'm not sure that this addresses your question.  If it doesn't, here's a shorter answer: It is part of normal life; it isn't a problem; and we all share it to some degree or other.
1153023
It is really surprising that there are still people who think they have a PROOF for statements like 1=2, 1=0, etc.\nFirst they should UNDERSTAND what is a PROOF.\n\nPROOF is a sequence of logically correct steps. If ONE of your step is wrong then it is NOT called a proof.\n\nI think these people should go back to Primary schools and LEARN the fundamentals.
1041583
Godel's theorem is a rather technical result about axiom systems. It is widely misunderstood, so be careful of popular explanations of it.\n\nThe set-up is as follows:\n\nAn axiom system consists of a collection of symbols and a collection of axioms (which are strings of symbols) and rules of deduction (which are ways of taking strings of symbols and obtaining another string of symbols). For Godel's result, there should only be countably many symbols (usually there are only finitely many in practice) and it should be possible to decide whether a string is an axiom via a computer program. There are also rule for deciding which strings are 'well formed'.\n\nMost of mathematics can be done in the context of axiom systems of this sort. For example, Zormelo-Fraenkl set theory is of this type.\n\nGodel's result says that for any axiom system which is strong enough to model the natural numbers there is a well formed formula which can neither be proved nor disproved from the axiom system.\n\nEssentially, Godel's result shows that there is no reasonable axiom system for the natural numbers that can answer every question that arises in that system. For example, with set theory, it is known that the axiom of choice can neither be proved nor disproved from just the ZF axioms. If the axiom of choice is added, it is known that the continuum hypothesis can neither be proved nor disproved from the new axiom system.\n\nProbably, the best way to interpret Godel's result is to say that axiom systems are inherently limited in their ability to answer questions that arise in those systems. If they are strong enough, there will be questions that can't be answered in the system. That doesn't mean the questions are unanswerable because considerations outside the system (aesthetics, for example) may dictate an answer.