Real Analysis Proof Editing review and Explanation

[DutifulJaguar9]

This is the question "Prove the following by using: a direct argument with two cases for n; and the properties of even and odd numbers.
[MATH]\forall n\in\mathbb{N}, n^{3}+n+2 [/MATH] is even."

I understand what I need to do to show that this is true. I show that:

If n is odd then n³ is odd, n is odd, 2 is even, then n³+n+2 is even

& If n is even, then n³is even, n is even, 2 is even, n³+n+2 is even
Both because of the properties of even and odd numbers.

However, I don't know how to show that in a formal proof (I also don't think I understand what a direct argument is.
Thanks

 

[ksdhart2]

Exact usage of terminology varies from class to class and from textbook to textbook, but I interpret "a direct argument with two cases for n" as meaning exactly what you did - first assume \(n\) is odd and show the proposition holds, then assume \(n\) is even and show the proposition holds for that case too.

The only way I can think of to make it more formal would be to explicitly state \(\text{Let } n = 2k + 1, \: k \in \mathbb{Z}\) and \(\text{Let } n = 2k, \: k \in \mathbb{Z}\), but I don't think that would really be necessary.

 

[DutifulJaguar9]

Exact usage of terminology varies from class to class and from textbook to textbook, but I interpret "a direct argument with two cases for n" as meaning exactly what you did - first assume \(n\) is odd and show the proposition holds, then assume \(n\) is even and show the proposition holds for that case too.

The only way I can think of to make it more formal would be to explicitly state \(\text{Let } n = 2k + 1, \: k \in \mathbb{Z}\) and \(\text{Let } n = 2k, \: k \in \mathbb{Z}\), but I don't think that would really be necessary.

Thank you. I think that I understand. When it says "Prove the following by using" does that mean make a two-column proof?

 

[Steven G]

This is the question "Prove the following by using: a direct argument with two cases for n; and the properties of even and odd numbers.
[MATH]\forall n\in\mathbb{N}, n^{3}+n+2 [/MATH] is even."

I understand what I need to do to show that this is true. I show that:

If n is odd then n³ is odd, n is odd, 2 is even, then n³+n+2 is even

& If n is even, then n³is even, n is even, 2 is even, n³+n+2 is even
Both because of the properties of even and odd numbers.

However, I don't know how to show that in a formal proof (I also don't think I understand what a direct argument is.
Thanks

Hi,
Your proof is perfectly fine. What do you not like about it?

 

[ksdhart2]

...does that mean make a two-column proof?

You could do this, if you wish. Whether or not you should depends a great deal on what your teacher/instructor/professor wants. If you have any similar examples from your textbook and/or that were worked in-class, or any similar examples of proofs that were graded satisfactorily, I'd follow those. As it stands though, if I were grading this proof, I would award you full marks.

 

[Steven G]

Thank you. I think that I understand. When it says "Prove the following by using" does that mean make a two-column proof?

No, it does NOT mean to do a two column proof. When it said to use two cases that is just what you did!. One case was for when was even and the 2nd case was for n odd. A direct proof means (to me) not a proof by contradiction. So what is a proof by contradiction? You suppose that the theorem is wrong and then get a contradiction showing that the theorem is valid

 

[pka]

Thank you. I think that I understand. When it says "Prove the following by using" does that mean make a two-column proof?

Does anyone still require a two-column proof? I sincerely hope not.
Mathematical proofs should be written in paragraph form. That means using clear and complete well-formed sentences.

 

[HallsofIvy]

The only places I have ever seen "two column proofs" are in high-school geometry classes!