Help with a logical derivation of set theoretical statement

In summary, the conversation discusses the proof of Dedekind's cut in Rudin's Principles of Mathematical Analysis. The conversation mentions the definition of \alpha and \beta, and a statement that needs to be derived. It is explained that q\in\alpha means q is a member of \alpha, and q\in\beta means there exists some r>0 such that -q-r\notin\alpha. It is then shown that if q\in\alpha, then q\notin\beta, which helps to understand the relationship between \alpha and \beta.
  • #1
julypraise
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0

Homework Statement


This is actually from the proof of Dedekind's cut in Rudin's Principles of Mathematical analysis on the page 19. It says when [tex]\alpha\in\mathbb{R}[/tex] ([tex]\alpha[/tex] is a cut) is fixed, [tex]\beta[/tex] is the set of all [tex]p[/tex] with the following property:

There exists [tex]r>0[/tex] such that [tex]-p-r\notin\alpha[/tex].​

From the given above, I need to derive that

if [tex]q\in\alpha[/tex], then [tex]q\notin\beta[/tex].​

But I cannot reach this statement as my explanation for this is in the below.

Homework Equations





The Attempt at a Solution


The draft I have done so far is that, as defining [tex]\beta[/tex] such that

[tex]\beta=\left\{p|\exists r\in\mathbb{Q} (r>0 \wedge -p-r \notin \alpha)\right\}[/tex],​

I derived

[tex]p \notin \beta \leftrightarrow
\forall r \in \mathbb{Q} (r>0 \to -p-r\in\alpha)[/tex].​

And I'm stucked here. From the last statement, I cannot derive the conclusion I was meant to derive. If anyone gives me help, I will give thanks.
 
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  • #2


Thank you for bringing this up. It seems like you are on the right track, but let me try to explain it in a different way.

First, let's recall the definition of \beta as given in the post: \beta is the set of all p such that there exists an r>0 such that -p-r\notin\alpha.

Now, let's consider the statement q\in\alpha. This means that q is a member of the cut \alpha, which is a set of all rational numbers less than some real number.

If q\in\alpha, then by definition of \alpha, there exists some r>0 such that q+r\in\alpha.

But wait, what does this mean? This means that q+r is a rational number that is less than some real number, which is the definition of being in a cut.

Now, let's consider the statement q\in\beta. This means that there exists some r>0 such that -q-r\notin\alpha.

But we just showed that q+r is a rational number that is less than some real number, which means that -q-r cannot be in \alpha.

Therefore, if q\in\alpha, then q\notin\beta.

I hope this helps you understand the relationship between \alpha and \beta. Keep up the good work in studying Dedekind's cut!
 

1. What is a logical derivation?

A logical derivation is a step-by-step process of using logical rules and principles to reach a conclusion from given premises. It is a way of logically proving the validity of a statement or argument.

2. What is a set theoretical statement?

A set theoretical statement is a statement that describes the relationships between different sets and their elements. It is a mathematical statement that uses symbols and logical operators to represent sets and their properties.

3. How do you create a logical derivation for a set theoretical statement?

To create a logical derivation for a set theoretical statement, you need to start with the given premises and use logical rules and principles to make deductions and reach a conclusion. This process involves breaking down the statement into smaller, more manageable parts and using logical reasoning to connect them.

4. What are some common logical rules used in set theoretical derivations?

Some common logical rules used in set theoretical derivations include the laws of sets (such as the commutative, associative, and distributive laws), the laws of logic (such as the law of identity and the law of non-contradiction), and the rules of inference (such as modus ponens and modus tollens).

5. Why is it important to use logical derivations in set theory?

Using logical derivations in set theory helps to ensure that our conclusions are valid and logically sound. It allows us to prove the relationships between sets and their elements, and to make inferences and deductions based on given premises. This helps us to understand and analyze complex mathematical concepts and to make accurate and precise statements about sets and their properties.

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