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Establish a [itex]\bullet[/itex] 0 = 0

  1. Aug 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Establish a [itex]\bullet[/itex] 0 = 0

    2. Relevant equations

    A few axioms I thought to be relevant..

    3. The attempt at a solution

    a = a

    a [itex]\bullet[/itex] 0 = 0 [itex]\bullet[/itex] a

    (a [itex]\bullet[/itex] 0) - (0 [itex]\bullet[/itex] a) = 0

    a(1 [itex]\bullet[/itex] 0) - a(0 [itex]\bullet[/itex] 1) = 0

    a[(1 [itex]\bullet[/itex] 0) - (0 [itex]\bullet[/itex] 1)] = 0

    a [itex]\bullet[/itex] 0 = 0
     
  2. jcsd
  3. Aug 23, 2012 #2

    Mark44

    Staff: Mentor

    What set does a belong to? What operation does ##\bullet ## represent?

    What is the actual problem statement?
     
  4. Aug 23, 2012 #3
    What we know is a is an element of some field F, and the dot is multiplication. What I wrote as the problem is the entire question.
     
    Last edited: Aug 23, 2012
  5. Aug 23, 2012 #4

    Mark44

    Staff: Mentor

    "element of some field F" wasn't in the original post.

    Every element in a field has an additive identity, and every nonzero element has a multiplicative identity.
     
  6. Aug 23, 2012 #5

    Zondrina

    User Avatar
    Homework Helper

    So you want to prove :

    a0 = 0 in some field F.

    So I'm presuming you're allowed to assume the existence of a unique zero element and unique additive inverses.

    Then :

    a0 = 0 + a0
    = (-(a0) + a0) + a0
    .....

    As much as id like to help you more, the rest should be obvious. Just use the axioms to justify your steps.
     
  7. Aug 24, 2012 #6

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    A similar approach is to use that 0=0+0.

    In this attempt, you haven't made it clear how the first equality implies the second.
     
  8. Aug 24, 2012 #7
    [itex]\bullet[/itex] is a label for the operation of multiplication in a field F. There is an operation of addition in the field, for which the following is true:
    [tex]
    x = x + 0, \forall x \in F
    [/tex]
    Multiply by a, and use the distributive law:
    [tex]
    a \bullet x = a \bullet (x + 0)
    [/tex]
    [tex]
    a \bullet x = a \bullet x + a \bullet 0
    [/tex]
    Use the group properties of addition of the field. What can you say about [itex]a \bullet 0[/itex] then? Similarly, you can multiply from the left and draw a conclusion for [itex]0 \bullet a[/itex].
     
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