is that proof correct?

[triclino]

To prove : for all natural Nos ,n : [math]n^2 [/math] is even[math]\Longleftrightarrow[/math] n is even.....................A

 

The following proof was suggested:

 

for all ,nεN :

 

n=2k or n= 2k + 1 ......κεN

 

But ,[math] n^2 = (2k + 1)^2 =4k^2 + 4k +1 = 2(2k^2 +2k) + 1[/math]

 

Hence : [math]n^2[/math] is not even or n is even....................1

 

Also:

 

[math] n^2 =(2k)^2 = 2(2k^2)[/math]

 

Hence: [math]n^2[/math] is even or n is not even......................2

 

 

From (1) and (2) we conclude: (A)

[timo]

It is confusing to say the least. For example: Do you think that anyone not willing to spend a relatively long time de-cyphering what you wrote understands what "for all ,nN:" means? You probably mean "for all n in N" or even simpler "for all natural numbers n", right? Then, technically a proof for "A <=> B" usually consists of two steps: a) show that A=>B, b) show that B=>A. I am not even sure if that is the path you take here.

 

Bottom line: I don't get what you are writing there. At least to some extent it is because your presentation is sloppy. Don't be afraid to use words.

Edited November 4, 2009 by timo

[the tree]

@triclino

Yes it is. Although maybe part (1) should state that (2k²+2k) is in N, just so your point is obvious.

 

@timo

Maybe your browser isn't rendering the character 'ε' but it is definitely there. Aside from that there is nothing wrong with their presentation. Don't be harsh just for the sake of it.

[timo]

Must have been my browser, yes. It now shows up.

 

I am not harsh for the sake of it, I say that I found the proof badly readable. This might just be my personal impression, strongly influenced by missing characters of course. But there also is the "or" which must be read as an exclusive or for the statement to still make any sense and still not being very elegant. If you're fine with it, then that's a good sign. To quote an old math prof of mine: "A proof is an argument that people understand and agree with."

 

It might admittedly not have been Triclino's question whether the presentation is ok (although no one really expect something miraculous to happen in the proof, I guess).

 

So -also in the light of the missing symbols now showing up- let me rephrase my initial post: It is not complete crap. But if I were given this as a homework solution (without missing characters, of course) I would not give full points (except perhaps one of one).