Prove Rational Numbers Have Midpoint: x,y ∈ Q

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In summary, 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.
  • #1
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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  • #2
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.
 
  • #3
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)?
 
  • #4
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.
 
  • #5
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.
 
  • #6
Ah ok i see , thanks guys.
 

FAQ: Prove Rational Numbers Have Midpoint: x,y ∈ Q

1. What are rational numbers?

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not equal to zero. They can be written in the form of a/b, where a and b are integers.

2. What is a midpoint?

A midpoint is the exact middle point between two numbers. It is the point that divides a line segment into two equal parts.

3. How can we prove that rational numbers have a midpoint?

We can prove that rational numbers have a midpoint by using the definition of a midpoint, which is the point that divides a line segment into two equal parts. We can show that for any two rational numbers, we can find a rational number that is exactly in the middle of them.

4. Can irrational numbers have a midpoint?

No, irrational numbers cannot have a midpoint. This is because irrational numbers cannot be expressed as a ratio of two integers, so there is no way to find a rational number that is exactly in the middle of two irrational numbers.

5. Why is it important to prove that rational numbers have a midpoint?

It is important to prove that rational numbers have a midpoint because it helps us understand the properties of rational numbers and how they behave. It also allows us to use rational numbers in various mathematical calculations and applications, such as finding the average of a set of numbers.

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