How to express this statement using quantifiers

  • #1
71
2
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.
 

Answers and Replies

  • #2
Erland
Science Advisor
738
136
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.
 
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  • #3
71
2
Thank you! That makes sense!
 

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