# Closed non-commutative operation on N

1. Mar 25, 2009

### tarkimos

The problem statement, all variables and given/known data

(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

Last edited: Mar 25, 2009
2. Mar 25, 2009

### 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).

3. Mar 25, 2009

### tarkimos

Okay, that clears some things up, thanks.
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?

4. Mar 25, 2009

### HallsofIvy

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

5. Mar 25, 2009

### tarkimos

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

6. Mar 26, 2009

### CompuChip

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