MATH220 April 2005
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[hide]Question 03 (c)
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What does "Cantor's Diagonalization Argument" prove? (You don't have to describe the proof itself - just what it proves.)
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Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
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[show]Hint
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Here the question simply aims at having you write a very short paragraph about what Cantor's argument proves, no need for complicated math, just a sentence or two that explains what he proved with that argument. (Yes, that's math too :) )
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Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
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[show]Solution
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Cantor's Diagonalization Argument is used to prove that the set of all real numbers is uncountable.
More precisely, it says that if you claim that the set of all real numbers is actually countable, that is, you have a bijection between all real number and the natural numbers, then he would be able to exhibit you a real number (using the diagonalization argument) that is not listed by your bijection. This means it is a contradiction to assume that such a bijection could exist and hence the real numbers are uncountable.
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