Is the set of integers Z={0,+-1,+-2, } .

[florenti]

Is the set of integers Z={0,+-1,+-2,...}...

Hi,

Can anybody help me.

I know that integers under Addition is a group, but

Is the set of Integers Z={0,+-1,+-2,...} together with the operation of subtraction a (noncommutative) group.

Thanks a lot

Flor

 

[HallsofIvy]

Well, have you tried anything at all?

In order to be a group, a set with an operation must satisfy these rules:

1) There is an "identity". Here that means there must be some integer, e, such that x- e= x and e- x= x for any integer x. Is there such a number?

2) If there is an identity, e, then every integer must have an "inverse", "-x", such that x- (-x)= e. Is there such a number?

3) The operation must be associative. For any three integers a, b, c, (a-b)- c must be equal to a-(b-c). Is that true?

 

[tiny-tim]

Welcome to PF!

I know that integers under Addition is a group, but

Is the set of Integers Z={0,+-1,+-2,...} together with the operation of subtraction a (noncommutative) group.

Hi Flor! Welcome to PF!

A group must have an inverse of every element, so the non-negative integers are not a group (I think they're a "semi-group").

The integers Z={0,+-1,+-2,...} are a commutative group …

commutative just means that, for example, 7 + 11 = 11 + 7.

(and you don't need to mention the operation of subtraction … subtraction is the inverse of addition, so it has to be in the group anyway.)

 

[florenti]

Thanks a lot,

So the set of integers with operation of subtraction is not a group because the subtraction of integers is not associative.So a-(b-c) is not equal to (a-b)-c.

Thanks a lot,

florent