1. ## dedekind sets

How do you prove sqrt(2) > 1 using dedekind sets

2. Originally Posted by l888l888l888
How do you prove sqrt(2) > 1 using dedekind sets
I must wonder if by "using dedekind sets" you mean Dedekind cuts?

3. yes my professor uses both in his lecture. sorry

4. Originally Posted by l888l888l888
How do you prove sqrt(2) > 1 using dedekind sets
What is the definition of "1" as a Dedekind cut? What is the definition of $\displaystyle \sqrt{2}$ as a Dedekind cut? What does ">" mean for Dedekind cuts?

5. for sqrt(2) it is defined as :{x belongs to Q: (X^2) < 2 V X < 0}
for 1 it is defined as : {x belong to Q: X<1}
tO PROVE > you have to prove {x belongs to Q: X<1} IS A SUBSST OF
{X belongs to Q: (x^2) < 2 V X < 0}

6. Originally Posted by l888l888l888
for sqrt(2) it is defined as :{x belongs to Q: (X^2) < 2 V X < 0}
for 1 it is defined as : {x belong to Q: X<1}
tO PROVE > you have to prove {x belongs to Q: X<1} IS A SUBSST OF
{X belongs to Q: (x^2) < 2 V X < 0}
And that is straight forward, isn't it? If x is in the Dedekind cut "1" then either x< 0 or $\displaystyle 0\le x< 1$. In the first case, x in the Dedekind cut "$\displaystyle \sqrt{2}$" because that cut includes "X< 0". In the second case, $\displaystyle 0\le x< 1$, what can you say about $\displaystyle x^2$?

7. x^2 is also between 0 and 1