Question

Is the set of positive integers an example of an abelian group?

Answer 1

if not what would be an example?

Answer 2

you cannot ask such a question without specifying what the operation is

Answer 3

is it \(\{\mathbb{N},+\}\) ?

Answer 4

or perhaps \(\{\mathbb{N},\times\}\)

Answer 5

the answer is "no" in any case, but it makes a difference for the reasoning

Answer 6

the first does not have an identity element so the answer is no for that reason

Answer 7

okay... why is it not.... and what would be an example of one?

Answer 8

ummm what about R,+

Answer 9

every group must have an identity element for the given operation
the set of natural numbers under addition does not, since it does not include zero

Answer 10

if you include zero, it still is not a group, because there also must be inverses, and you cannot solve \(2+n=0\) for \(n\) if \(n\in \mathbb{N}\)

Answer 11

if you include all integers, then it would be a group under addition

Answer 12

Ok ... so {Z,+}

Answer 13

that would be an example, yes
there are countless examples

Answer 14

okay, so in order to determine what one would be.... then you take the set... and the operation... and pick any a,b,c and make sure that it is
1. closure
2. associative
3. identity
4. inverse
5. communitive

Answer 15

those are the properties for an abelian group, yes

Answer 16

thank you.

Answer 17

yw

Answer 18

and yes, \(\{\mathbb{R},+\}\) is an abelian group, however \(\{\mathbb{R},\times \}\) is not