Closed non-commutative operation on N

[tarkimos]

Homework Statement

(i) Give an example of a closed non-commutative binary operation on N (the set of all natural numbers).
(ii) Give an example of a closed non-associative binary operation on N.

The attempt at a solution
This has me stumped, there must be something simple that I'm missing. I was thinking divisibility ('|')...

EDIT: Looking back at my first topic, wow, I've come a long way since when I last posted

 

[CompuChip]

Note that a binary operation goes from N x N into N again. Divisibility is therefore not a really good example, for example, which number is 3 | 6? What does (3 | 6) | 4 mean?

Instead, try something simpler. I think you can even use the same counterexample for both. Let me give you a hint:
3 - 5 = - (5 - 3).

 

[tarkimos]

Note that a binary operation goes from N x N into N again. Divisibility is therefore not a really good example, for example, which number is 3 | 6? What does (3 | 6) | 4 mean?

Okay, that clears some things up, thanks.

Instead, try something simpler. I think you can even use the same counterexample for both. Let me give you a hint:
3 - 5 = - (5 - 3).

I need to use a counter-example?
I still can't work it out, sorry. I think that your hint went clear over my head.

I can't use subtraction (as it is not a closed operation in N), can I?

 

[HallsofIvy]

Then define a new operation. For example, x*y= 2x+ y is clearly non-commutative.

 

[tarkimos]

Then define a new operation. For example, x*y= 2x+ y is clearly non-commutative.

Ah, thank you very much. I wasn't looking at the questions broadly enough. :)

 

[CompuChip]

Sorry, I meant example instead of counterexample.
And I thought it said Z in which N is indeed closed.
My apologies.