귀류법

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  1. Beyond that, all I can do is offer some advice about structuring a proof by contradiction.[1]
  2. The tricky thing is that it is hard to test statements within a proof by contradiction for plausibility.[1]
  3. To avoid this issue, it's often best to try to build up as much of the framework for the proof as possible outside of the actual proof by contradiction.[1]
  4. If you make the actual proof by contradiction as short and simple as possible, then it will help you avoid mistakes and help other people check your work.[1]
  5. Certain types of proof come up again and again in all areas of mathematics, one of which is proof by contradiction.[2]
  6. You may like to try this challenge which involves a slightly different proof by contradiction to prove the same result.[2]
  7. This would have been much quicker than going through the whole proof by contradiction.[2]
  8. This is a basic rule of logic, and proof by contradiction depends upon it.[3]
  9. It is that last condition of truth and falsity that is exploited, powerfully and universally, by proof by contradiction.[3]
  10. That alerts the reader that you are using proof by contradiction and will plug away at the proof until it collapses logically.[3]
  11. You may well benefit from rereading it several times, but once you do, you should feel more confident in your understanding of proof by contradiction.[3]
  12. Another method of proof that is frequently used in mathematics is a proof by contradiction.[4]
  13. So if we want to prove a statement \(X\) using a proof by contradiction, we assume that \(\urcorner X\) is true and show that this leads to a contradiction.[4]
  14. When we try to prove the conditional statement, “If \(P\) then \(Q\)” using a proof by contradiction, we must assume that \(P \to Q\) is false and show that this leads to a contradiction.[4]
  15. To begin a proof by contradiction for this statement, we need to assume the negation of the statement.[4]
  16. In a proof by contradiction, it is shown that the denial of the statement being proved results in such a contradiction.[5]
  17. This fact is used in proof by contradiction.[5]
  18. An existence proof by contradiction assumes that some object doesn't exist, and then proves that this would lead to a contradiction; thus, such an object must exist.[5]
  19. Proof by contradiction also depends on the law of the excluded middle, also first formulated by Aristotle.[5]
  20. The steps taken for a proof by contradiction (also called indirect proof) are: Assume the opposite of your conclusion.[6]
  21. An indirect proof establishes that the opposite conclusion is not consistent with the premise and that, therefore, the original conclusion must be true.[6]
  22. Proof by contradiction is used frequently in classical mathematics.[7]
  23. The approach of proof by contradiction is simple yet its consequence and result are remarkable.[8]
  24. Just a word of advice, use the proof by contradiction if the methods of direct and contrapositive proofs do seem to fail.[8]
  25. Having said this, I should note that it is considered bad style to write a proof by contradiction when you can give a direct proof.[9]
  26. In those situations, the proof by contradiction often looks awkward.[9]
  27. In some cases, proof by contradiction is used as part of a larger proof --- for instance, to eliminate certain possibilities.[9]
  28. This is of course the classic proof by contradiction, and one can even give a kind of quasi-proof that it must use contradiction.[10]
  29. At first sight, this looks like a direct argument rather than a proof by contradiction: we used the hypothesis that to deduce that has a property that obviously implies irrationality.[10]
  30. It’s obvious that a rational number has a terminating continued fraction, because as you work it out the denominators keep decreasing … oops, sorry, that was a proof by contradiction.[10]
  31. So perhaps the answer is indeed that if you are trying to prove a negative statement, then you have to use a proof by contradiction.[10]
  32. The method of proof by contradiction is to assume that a statement is not true and then to show that that assumption leads to a contradiction.[11]
  33. For many students, the method of proof by contradiction is a tremendous gift and a trojan horse, both of which follow from how strong the method is.[11]
  34. They are closely related, even interchangeable in some circumstances, though proof by contradiction is more powerful.[12]
  35. Proof by contradiction makes some people uneasy—it seems a little like magic, perhaps because throughout the proof we appear to be `proving' false statements.[12]
  36. Still, there seems to be no way to avoid proof by contradiction.[12]
  37. Are there any clues that might lead you to think that an indirect proof might be a good idea?[12]
  38. And that would be your proof by contradiction.[13]
  39. Famous results which utilized proof by contradiction include the irrationality of and the infinitude of primes.[14]
  40. This technique usually works well on problems where not a lot of information is known, and thus we can create some using proof by contradiction.[14]
  41. Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem.[15]
  42. In an indirect proof of a theorem P → Q, you assume P → Q and ~ Q, and try to show ~ P. You can write this as ((P → Q) ∧ ~ Q) →~ P, and construct a truth table for this compound statement.[16]
  43. The key to a proof by contradiction is that you assume the negation of the conclusion and contradict any of your definitions, postulates, theorems, or assumptions.[16]
  44. Hypothesis testing can be thought of as being the probabilistic counterpart of proof by contradiction.[17]
  45. When we use proof by contradiction, we start by assuming the opposite of A (denoted ~A or not A) as being true, and then we try to reach to a contradiction (a statement that is always false).[17]
  46. As in the case of proof by contradiction, just because we may not have a good idea of how to reach a contradiction once we assumed ~A to be true, this does not necessarily mean that ~A is true.[17]
  47. In fact, we can go in the other direction, and use the law of the excluded middle to justify proof by contradiction.[18]
  48. If all else fails, use a proof by contradiction.[18]
  49. Sometimes a proof by contradiction is necessary, but when it isn’t, it can be less informative than a direct proof.[18]
  50. To prove the statement “A implies B,” a proof by contradiction assumes that both A and “not B” are true, and then shows that this is impossible.[19]
  51. Another commonly given example of proof by contradiction is Cantor’s diagonalization argument showing that the set of real numbers is “bigger” than the set of counting numbers.[19]
  52. But by trying a proof by contradiction, we have in some sense a tangible "villain" to fight against.[19]
  53. Proof by contradiction is a strong method of proving statements.[20]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Dangers of proof by contradiction
  2. ↑ 2.0 2.1 2.2 An Introduction to Proof by Contradiction
  3. ↑ 3.0 3.1 3.2 3.3 Proof by Contradiction (Definition, Examples, & Video)
  4. ↑ 4.0 4.1 4.2 4.3 3.3: Proof by Contradiction
  5. ↑ 5.0 5.1 5.2 5.3 Proof by contradiction
  6. ↑ 6.0 6.1 Making Mathematics: Mathematics Tools: Proof by Contradiction
  7. ↑ proof by contradiction in nLab
  8. ↑ 8.0 8.1 Proof by Contradiction
  9. ↑ 9.0 9.1 9.2 Proof by Contradiction
  10. ↑ 10.0 10.1 10.2 10.3 When is proof by contradiction necessary?
  11. ↑ 11.0 11.1 Mathematical Proof/Methods of Proof/Proof by Contradiction
  12. ↑ 12.0 12.1 12.2 12.3 2.6 Indirect Proof
  13. ↑ CA Geometry: Proof by contradiction (video)
  14. ↑ 14.0 14.1 Art of Problem Solving
  15. ↑ proof by contradiction
  16. ↑ 16.0 16.1 Geometry: Proof by Contradiction: The Advantage of Being Indirect
  17. ↑ 17.0 17.1 17.2 The Link Between Hypothesis Testing and Proof by Contradiction
  18. ↑ 18.0 18.1 18.2 5. Classical Reasoning — Logic and Proof 3.18.4 documentation
  19. ↑ 19.0 19.1 19.2 In Praise of Proofs by Contradiction that Aren't
  20. ↑ Methods and Techniques for Proving Inequalities

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  • [{'LOWER': 'indirect'}, {'LEMMA': 'proof'}]
  • [{'LOWER': 'apagogical'}, {'LEMMA': 'argument'}]
  • [{'LOWER': 'proof'}, {'LOWER': 'by'}, {'LOWER': 'assuming'}, {'LOWER': 'the'}, {'LEMMA': 'opposite'}]
  • [{'LOWER': 'reductio'}, {'LOWER': 'ad'}, {'LEMMA': 'impossibilem'}]
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