Spivak Calc: Is my proof correct?

In summary, the given statements P10-P12 outline the Trichotomy law and the properties of the collection P, which is defined as the set of positive numbers. P10 states that for any number a, one and only one of the three options holds true: a = 0, a is in P, or -a is in P. P11 states that if a and b are in P, then their sum is also in P. Similarly, P12 states that if a and b are in P, then their product is also in P. Your attempts at solving P10-P12 demonstrate a clear understanding of the concepts and show how they can be applied in different scenarios.
  • #1
albert1993
18
0

Homework Statement


Prologue Chapter 1. Problem 8.

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(P10) Trichotomy law: For every number a, one and only one of the following holds:
(i) a = 0
(ii) a is in the collection P
(iii) -a is in the collection P
(P11) If a and b are in P, then a + b are in P
(P12) If a and b are in P, then a x b is in P.

The Attempt at a Solution



P10: Let (b-a) be any number d. Then P'10 reduces to
i) d = 0
ii) 0 <d
iii) 0 >d
This is in essence what P10 is, except with the variable being different

P11: Let there be numbers a, b, c, d.
If a<b and c<d, then by P'12 the following are true:
1) a+c < b+c
2) b+c < b+d
By P'11, a + c < b + d
Looking at the case where 0 < b and 0 < d, b + d > 0 which is P11.

P12: If 0 < a and 0 < c, by P'13, 0 < ac, which is P12
 
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  • #2
.Great job recognizing the similarities between the given statements and your attempts at a solution! Here are some additional thoughts and suggestions for your response:

- For P10, it may be helpful to clarify that "P" refers to the collection of positive numbers. This can help make the statement more clear and specific.
- In your attempt at P10, it may also be helpful to explain how you arrived at the three possible cases for d. This can help the reader understand your thought process and reasoning.
- For P11, it may be helpful to explain why the statement "a + c < b + d" is true, based on the given statements P'12 and P'11. This can help strengthen your argument.
- In your attempt at P12, it may be helpful to clarify that "a" and "c" refer to positive numbers. This can help make the statement more clear and specific.
- Additionally, you may want to provide more detail or explanation for how you arrived at the statement "0 < ac" from P'13. This can help the reader understand your reasoning and strengthen your argument.

Overall, your response shows good understanding of the given statements and how they relate to each other. Just remember to provide enough detail and explanation to make your argument clear and convincing. Great job!
 

1. What is "Spivak Calc"?

"Spivak Calc" refers to the textbook "Calculus" written by Michael Spivak. It is a comprehensive and rigorous introduction to calculus, often used in advanced undergraduate courses.

2. How do I know if my proof is correct?

To determine the correctness of a proof in "Spivak Calc", it is important to carefully follow the logic and steps presented in the textbook. Additionally, it is helpful to consult with a teacher or peer for feedback and verification.

3. Can I use alternative methods to prove a theorem in "Spivak Calc"?

While "Spivak Calc" presents specific methods for proving theorems, it is possible to use alternative methods as long as they follow the same logical steps and lead to the same conclusion.

4. Are there any common mistakes to avoid when writing a proof in "Spivak Calc"?

Some common mistakes to avoid when writing a proof in "Spivak Calc" include not fully understanding the definitions and concepts presented in the textbook, making incorrect assumptions or skipping steps in the proof, and not providing enough explanation and justification for each step.

5. How can I improve my proof-writing skills in "Spivak Calc"?

To improve proof-writing skills in "Spivak Calc", it is important to carefully read and understand the material, practice writing proofs regularly, and seek feedback and guidance from teachers or peers. It is also helpful to study and analyze well-written proofs from the textbook or other sources.

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