• May 2nd 2010, 12:43 PM
MissMousey
Find a subset of Z that is closed under addition but is not a group of the additive group Z.

Will the set of natural number be suffice?
• May 2nd 2010, 09:00 PM
Drexel28
Quote:

Originally Posted by MissMousey
Find a subset of Z that is closed under addition but is not a group of the additive group Z.

Will the set of natural number be suffice?

Well, that depends how you define the naturals. Why do you think that will work?
• May 3rd 2010, 02:32 AM
MissMousey
Quote:

Originally Posted by Drexel28
Well, that depends how you define the naturals. Why do you think that will work?

Let S be the set of even natural numbers where S = {2, 4, 6, ...}. This will be closed with respect to addition because the sum of two even natural numbers is an even natural number.

However, since the set does not contain inverses and does not contain the identity element, it does not satisfy group properties.
• May 3rd 2010, 02:39 AM
HallsofIvy
There are two commonly used definitions of the "natural numbers"- all positive integers or all non-negative integers. Either of those will work for this problem.