# Is my proof valid?

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.

## Answers and Replies

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
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 doesnt apply to any other rationals, so it doesnt really prove anything. Am i misinterpreting your statement ( im really bad at math so please excuse my lack of ability)?

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
If i construct a rational in between p/q and r/s , i doesnt apply to any other rationals, so it doesnt really prove anything. Am i misinterpreting your statement ( im 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.

Ibix
Science Advisor
2020 Award
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.