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Does this small odd and even proof works?

  1. Aug 28, 2013 #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.
     
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  3. Aug 28, 2013 #2

    Office_Shredder

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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. Aug 28, 2013 #3
    Well ok then, I just want to know if that contradiction works.
     
  5. Aug 28, 2013 #4
    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. Aug 28, 2013 #5
    It is not proven but it is used in the definition that if a number is not even then it is odd.
     
  7. Aug 29, 2013 #6

    Office_Shredder

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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. Aug 29, 2013 #7
    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. Aug 29, 2013 #8

    Office_Shredder

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    That's a very nice proof
     
  10. Aug 29, 2013 #9
    I think I can at least feel this small 'beauty' feeling after sketching it. Thanks!
     
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