How to express this statement using quantifiers

[rasen58]

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.

 

[Erland]

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.

 

[rasen58]

Thank you! That makes sense!