Prove Rational Numbers Have Midpoint: x,y ∈ Q

  • Thread starter Thread starter kaos
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary
SUMMARY

This discussion centers on proving that between any two rational numbers, there exists a third rational number. The proof utilizes a contradiction approach, asserting that if no rational number exists between two given rationals, it leads to a logical fallacy. The participants emphasize the necessity of constructing a rational number between two arbitrary rationals, represented as p/q and r/s, to validate the claim universally. The conclusion is that a general construction method must be established to demonstrate the existence of a midpoint in the rational number set.

PREREQUISITES
  • Understanding of rational numbers (Q)
  • Familiarity with proof techniques, particularly proof by contradiction
  • Basic algebraic manipulation of fractions
  • Knowledge of logical quantifiers and their implications
NEXT STEPS
  • Learn how to construct a rational number between two given rational numbers using the formula (p/q + r/s) / 2
  • Study the properties of rational numbers and their density in the real number line
  • Explore advanced proof techniques in mathematics, focusing on contradiction and direct proofs
  • Investigate the implications of rational number properties in real analysis
USEFUL FOR

Mathematics students, educators, and anyone interested in understanding the properties of rational numbers and proof techniques in mathematical logic.

kaos
Messages
63
Reaction score
0
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.
 
Physics news on Phys.org
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.
 

Similar threads

Prove a rational fraction is equal to another
  • · Replies 5 ·
Replies
5
Views
2K
Prove that the equation is satisfied at least once
  • · Replies 1 ·
Replies
1
Views
2K
Proof given ##x < y < z## and a twice differentiable function
  • · Replies 8 ·
Replies
8
Views
2K
Prove if ##x<0## and ##y<z## then ##xy>xz## (Rudin)
  • · Replies 2 ·
Replies
2
Views
2K
For z = x+iy find the relationship between x and y
  • · Replies 2 ·
Replies
2
Views
2K
Prove that if x,y, and z are integers and
  • · Replies 24 ·
Replies
24
Views
3K
Find x,y,z in Sequence: 33, ?,?,?,88
  • · Replies 4 ·
Replies
4
Views
2K
Why Must n Equal 4q or 4q+2 If It Isn't 4q+1 or 4q+3?
  • · Replies 10 ·
Replies
10
Views
2K
Possible values an expression can take : ##\dfrac{x^2-x-6}{x-3}##
  • · Replies 7 ·
Replies
7
Views
2K
Prove abs(x)-abs(y) is less than or equal to abs(x-y)
  • · Replies 8 ·
Replies
8
Views
11K