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How to express this statement using quantifiers

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  1. Jan 14, 2016 #1
    There is a real number between any other two real numbers.
    I have two ways of writing it
    For all y and z there is some x such that y < x < z
    OR
    There is such an x such that for all y and z, y < x < z

    I'm confused as to which one is correct.
     
  2. jcsd
  3. Jan 15, 2016 #2

    Erland

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    Science Advisor

    None of them is fully correct, you must add the assumption that y < z, otherwise there would be a number x such that e.g. 2 < x < 1.

    If you add this assumption, the first one is correct, because x depends on y and z. If the second was correct, there would be an x which lies between any numbers y and z such that y < z, so there would be an x such that 0 < x < 1, 1 < x < 2, 55 < x < 971 etc. i.e. the same x would work for all y and z such that y < z.
     
  4. Feb 3, 2016 #3
    Thank you! That makes sense!
     
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