Does this small odd and even proof works?

  1. This is taken from Peter J. Eccles, Introduction to Mathematical Reasoning, page 17. This is not a homework because it is an example in the text. Prove that 101 is an odd number. The text has given a way of proving it and I just want to do it with my own approach.

    Prove that 101 is an odd number.

    Assume 101 is even. There is a number ##b## such that ##101=2b##. Adds 1 to both side. ##102=2b+1## The right side shouldn't be divisible by 2 but the left side can be divided by 2. A contradiction? Hence 101 is an odd number.
     
  2. jcsd
  3. Office_Shredder

    Office_Shredder 4,500
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    Given the level of the question it seems like you should prove that 102 can be divided by 2 (by stating what the multiplication is). At that point it's just as easy to say 101 = 50*2+1 so is odd but them's the shakes
     
  4. Well ok then, I just want to know if that contradiction works.
     
  5. Your proof also uses the fact that if a number isn't even, then it's odd. Again considering the level of the question, this fact may not have been proven yet.
     
  6. It is not proven but it is used in the definition that if a number is not even then it is odd.
     
  7. Office_Shredder

    Office_Shredder 4,500
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    Then you should probably think about proving that if a number is of the form 2b+1 then it's odd (in particular you have to prove it's not even)
     
  8. Prove that ##2b+1## is odd.

    Suppose ##2b+1## is even, then it exists an integer ##c##, where ##2b+1=2c##

    ##2c+1=2b+2## and ##2c-1 = 2b##, hence ##2b<2c<2b+2##. Further ##b<c<b+1##

    But there is no integer which is larger and smaller than the next consecutive integer. So ##2b+1## must be odd.

    I think this opens a new bag of cats, but at least I found this myself from scratch!

    (The book standard proof implicitly assumes this I believe)
     
    Last edited: Aug 29, 2013
  9. Office_Shredder

    Office_Shredder 4,500
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    That's a very nice proof
     
  10. I think I can at least feel this small 'beauty' feeling after sketching it. Thanks!
     
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