Prove Rational Numbers Have Midpoint: x,y ∈ Q

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The discussion revolves around proving that there exists a rational number between any two rational numbers, x and y. A proof by contradiction is presented, suggesting that if no such z exists, it leads to false conclusions about the relationships between x, y, and z. However, participants argue that the proof lacks validity since it does not specifically utilize properties of rational numbers. They recommend a straightforward construction method to find a rational number between two given rationals, p/q and r/s, emphasizing that this approach can be generalized. The conversation highlights the importance of clear mathematical reasoning and construction in proofs involving rational numbers.
kaos
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Can someone check if my proof is correct.Please exscuse the bad notation, I've no idea how to type the symbols.
The question was prove that between any 2 rational number , there is a third rational.

x,y ,z are elements of Q
(for all x ) (for all y) (there exist z)[x>z>y] <->
(for all x ) (for all y) (there exist z)[(x>z) ^ (z>y)]

Proof by contradiction:
Suppose its false that for any x and y , there exists a z between x and y

~((for all x ) (for all y) (there exist z)[x>z>y])
(there exists x) (there exists y)( for all z)[ (x< or = z) V (z < or = y)]
There is no x that is smaller than or equals to any z.
There is no y that is larger than or equals to any z.
Both are false, the disjunction is false.
Therefore the statement (there exists x) (there exists y)( for all z)[ (x< or = z) V (z < or = y)]
is false and the statement (for all x ) (for all y) (there exist z)[x>z>y] is true.
 
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Since you have not used any facts about rational numbers, it seems vanishingly unlikely that your proof is valid.
How about doing something really simple and obvious: given two rationals p/q and r/s construct a rational that lies between them.
 
If i construct a rational in between p/q and r/s , i doesn't apply to any other rationals, so it doesn't really prove anything. Am i misinterpreting your statement ( I am really bad at math so please excuse my lack of ability)?
 
kaos said:
If i construct a rational in between p/q and r/s , i doesn't apply to any other rationals, so it doesn't really prove anything. Am i misinterpreting your statement ( I am really bad at math so please excuse my lack of ability)?
P, q, r and s can be any integers (q, s nonzero). If you construct a rational between p/q and r/s then you will have provided a general construction for any given pair of rationals.
 
p/q and r/s are arbitrary rational numbers. Haruspex is suggesting that you construct an expression in terms of p, q, r and s that is rational and guaranteed to lie between the the two.
 
Ah ok i see , thanks guys.
 

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