Slight deviation of proof, would it be correct for integers? Review for exam

In summary, the conversation involved a discussion about changing parameters in proofs and testing out different cases. The person also shared an example of a proof and asked for feedback. The other person suggested using the fact that all integers are rational numbers in the proof, making it simpler. They also mentioned that they knew the person from attending the same university.
  • #1
mr_coffee
1,629
1
Hello everyone.

He told us he could of course change the parameters which he will of the proofs we have been working on so I'm testing out some cases but I want to make sure I'm doing it right.

Here is an example of a proof the boook had:
http://suprfile.com/src/1/3qaagxh/Untitled-1%20copy.jpg


Now where he has the statement: "If a and b are rational numbers..."
I'm changing that to:
"If a and b are integers..."
and now here is my proof, i think its correct but I have to make sure.

If a and b are integers, b != 0, and r is an irrational number, then a+ br is irrational.

Proof by Contradiction:

Suppose not. Suppose that a and b are integers, b != 0, and r is an irrational number such that a+br is rational. We must obtain a contradiction.

Since a, b are integers and a + br are rational, a+br = m/n for some integers m, n with n != 0.

Then

a + br = m/n

br = m/n - a
r = (m-an)/bn

where (m-an) and (bn) are integers since m, a, n, and b are integers, and bn is nonzero since b is nonzero. Therefore, r is rational, contradicting that r is irrational.

Thanks!
 
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  • #2
looks good to me. are you using the book my solow? btw, you you go to UPark?
 
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  • #3
I'm using Discrete Mathematics with appplications 3rd by Susanna S. Epp, and yeah i go to Penn State UPark. CSE260 is the class, hah how did you know.
 
  • #4
just a guess/facebook/google(thinkquest contest). many of my friend go there.
 
  • #5
yeah this place is pretty massive, well thanks for the help! I seem to post all questions but don't answer any hah. I did do thinkquest along time ago... I'm supprised that site is still up! well ttyl
 
  • #6
Since all integers are rational numbers, once you had proven it for a, b rational, exactly the same proof must hold for a, b integer. Actually, I wouldn't have written the rational numbers as fractions, i/j, etc. It is sufficient to use the fact that the rational numbers are closed under addition and multiplication:
Suppose a+ br, a, b rational b non-zero, is rational: a+ br= c where c is rational. Then br= c- a so r= (c-a)(1/b). Since b is a non-zero rational number 1/b is rational and so r= (c-a)(1/b) is rational- contradiction.
 
  • #7
That does seem a lot nicer, thanks!
 

1. What is a slight deviation of proof?

A slight deviation of proof refers to a minor change or adjustment made to a mathematical or scientific proof in order to achieve a desired result or to account for an unexpected outcome. It is often used to fine-tune a theory or hypothesis and can lead to new discoveries or insights.

2. Is it acceptable to have a slight deviation of proof when working with integers?

Yes, it is generally acceptable to have a slight deviation of proof when working with integers. Integers are whole numbers that can be positive, negative, or zero, and they are often used in mathematical proofs. As long as the deviation does not significantly alter the original proof or the validity of the final result, it is considered acceptable.

3. How can a slight deviation of proof affect the outcome of an exam?

A slight deviation of proof can potentially affect the outcome of an exam if it leads to an incorrect answer or a different conclusion than what was intended. However, if the deviation is acknowledged and explained, it can also demonstrate critical thinking and problem-solving skills, which may be beneficial for the exam.

4. Can a slight deviation of proof lead to a different solution for a problem involving integers?

Yes, a slight deviation of proof can potentially lead to a different solution for a problem involving integers. This is because even small changes in a proof can have a significant impact on the final outcome. However, it is important to carefully examine the deviation and its effects to determine if the new solution is valid.

5. How can one review for an exam that involves slight deviations of proof?

The best way to review for an exam that involves slight deviations of proof is to thoroughly understand the concepts and principles behind the proofs. This will allow you to recognize when a deviation is necessary and how it may affect the final result. Practice problems and discussions with peers or instructors can also help improve understanding and preparation for the exam.

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