Direct Proofs

[dayrinni]

I am taking a math course in Discrete math and I do not understand proofs. For example:

Prove if n is odd then n^2 is also odd.
So we assume n is odd, and the definition of an odd integer is 2k+1.
So now we have: n = 2k+1.
n^2 = (2k+1)^2
and then:
4k^2 +4k + 1

That is the whole proof, but I do not understand the connection of this. I know that we are replacing n with the definition of an odd integer, because we are assuming that it is true (as you do in direct proofs), and then we take that definition and place it into n^2. After expanding the square....I guess is where I am having the problem. I do not see how that is really proving anything. There is something (very easy I feel lol) that I am missing from this.

I also have another question: is it good to use rules of inference for this? I tried using proving this using rules of inference and I managed to prove it.

I need some more explaination here.

Thanks!

[Dr_Strangelove]

http://www.toe.sytes.net:65333/2l.jpg. Then 2l+1 is an odd integer if l is an integer.
I assume you have at your disposal the proof that the product of integers is an integer and the definiton of the square of an integer.

[fraleysinger]

Is this for orbital angular momentum?

[JasonRox]

I am taking a math course in Discrete math and I do not understand proofs. For example:

Prove if n is odd then n^2 is also odd.
So we assume n is odd, and the definition of an odd integer is 2k+1.
So now we have: n = 2k+1.
n^2 = (2k+1)^2
and then:
4k^2 +4k + 1

That is the whole proof, but I do not understand the connection of this. I know that we are replacing n with the definition of an odd integer, because we are assuming that it is true (as you do in direct proofs), and then we take that definition and place it into n^2. After expanding the square....I guess is where I am having the problem. I do not see how that is really proving anything. There is something (very easy I feel lol) that I am missing from this.

I also have another question: is it good to use rules of inference for this? I tried using proving this using rules of inference and I managed to prove it.

I need some more explaination here.

Thanks!

You got it all there.

Dr. Strangelove pointed exactly what you needed to finish it.

[HallsofIvy]

I am taking a math course in Discrete math and I do not understand proofs. For example:

Prove if n is odd then n^2 is also odd.
So we assume n is odd, and the definition of an odd integer is 2k+1.
So now we have: n = 2k+1.
n^2 = (2k+1)^2
and then:
4k^2 +4k + 1

That is the whole proof, but I do not understand the connection of this. I know that we are replacing n with the definition of an odd integer, because we are assuming that it is true (as you do in direct proofs), and then we take that definition and place it into n^2. After expanding the square....I guess is where I am having the problem. I do not see how that is really proving anything. There is something (very easy I feel lol) that I am missing from this.

I also have another question: is it good to use rules of inference for this? I tried using proving this using rules of inference and I managed to prove it.

I need some more explaination here.

Thanks!
There should have been one more step:
After
n^2= 4k^2 +4k + 1
write
n^2= 2(2k^2+ 2k)+ 1.

That shows immediately that n^2 is 2 times an integer (2k^2+ 2k), plus 1. Exactly what you said was required for an odd number.