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

  1. Nov 10, 2008 #1
    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
     
  2. jcsd
  3. Nov 10, 2008 #2

    HallsofIvy

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

    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?
     
  4. Nov 10, 2008 #3

    tiny-tim

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    Welcome to PF!

    Hi Flor! Welcome to PF! :smile:

    A group must have an inverse of every element, so the non-negative integers are not a group (I think :redface: 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. :smile:

    (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.)
     
  5. Nov 10, 2008 #4
    Re: Is the set of integers Z={0,+-1,+-2,....}.....

    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
     
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