Help Solve Analysis Problem: Showing α is a Cut & αα=β

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In summary, the conversation is about a homework question asking for help with a proof involving the density of squares of rational numbers and cuts in the set of positive rational numbers. The person is also asking for guidance on how to include mathematical notation in their response.
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zmac
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Please help!
Here is a homework question that I need help starting!
It is a fact that the squares of the rational numbers are dense in the positive rational numbers, that is, for any r1,r2 in Q with 0< r1< r2, there exists p in Q with r1< p2<r2. Let β be a cut such that 0*< β. Let α be the subset of Q defined by α={p in Q: p< or =0 or p2 in β}. Use the abovementioned density of the squares of rationals to show that α is a cut and that αα=β.

ps How do I put mathematical notation here?

Any foresight you can give would be greatly appreciated!
Thanks
 
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  • #2
The usual questions: what have you tried, where are you stuck, what are the definitions, etc?
 
  • #3
I am having a little trouble interpreting the question. By p2, do you mean p squared?

Thanks,

Steve
 

1. What does it mean for α to be a cut?

In mathematics, a cut is a partition of a set into two disjoint subsets, such that every element in the first subset is less than every element in the second subset. In other words, α is a cut if it divides a set into two parts, with all elements in the first part being smaller than all elements in the second part.

2. How is αα=β related to the problem of showing α is a cut?

The equation αα=β, also known as the "cut property", is a fundamental property of cuts in mathematics. It states that if α is a cut, then the set of all elements less than α is equal to α itself. This property is important in proving that α is indeed a cut.

3. Why is it important to show that α is a cut?

Showing that α is a cut is important because it helps us understand the properties and behavior of cuts in mathematics. It also allows us to use cuts in various mathematical proofs and constructions.

4. How can we prove that α is a cut?

To prove that α is a cut, we must show that it satisfies the definition of a cut. This involves demonstrating that it divides a set into two disjoint subsets, with all elements in the first subset being less than all elements in the second subset. In addition, we must also show that αα=β holds, which is a crucial property of cuts.

5. Can you provide an example of α being a cut?

Yes, an example of α being a cut is the set of all rational numbers less than √2. This set can be divided into two subsets: the first subset contains all rational numbers less than √2, and the second subset contains all rational numbers greater than √2. This satisfies the definition of a cut, and we can also see that αα=β holds in this case.

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