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Does commutativity imply associativity?

  1. Jan 19, 2009 #1
    Hi everyone,

    Lately, I've been wondering whether commutativity implies associativity or not. (It's really hard for me to pay attention in my Calculus BC class.) I've never seen an example of a binary operation that is commutative but not associative. It seems intuitively true to me, but I don't know how to prove it (so maybe it's not true?). I also can't find anything on the web about this.

    Can anyone help? Thanks!
  2. jcsd
  3. Jan 19, 2009 #2
    Given ab = bc for all a, b, can you prove (ab)c = a(bc) for all a, b, c? Manipulate these equations for a while.

    To think of examples, try defining specific binary operations on small sets like {0, 1}.

    For an answer that's sort of intuitive, search Wikipedia for "commutative non-associative magma".
  4. Jan 19, 2009 #3


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    Homework Helper

    Never played rock paper scissors?



  5. Jan 20, 2009 #4
    Thanks for the answers. :) It never came across me to define an operation myself but that rock paper scissors example really helped. (Algebra seems so much more complicated now!!!)
  6. Jan 20, 2009 #5


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    [corected I had RS=S before clearly false]
  7. Jan 21, 2009 #6
    When I was still in school I wanted to (falsely) calculate my average grade in math as following:

    First method
    I wrote three exams with marks 1,2 and 4. I would take the average of 1 and 2, so I got a 1.5. Then I would take 1.5 and form the average with 4:

    (1+2)/2 = 1.5
    (1.5+4)/2 = 2.75

    Second method
    However, had I taken another order the result was different:
    I would take 2 and 4 and form the average, so I got a 3. Then I would take 3 and form the average with 1:

    (2+4)/2 = 3
    (3+1)/2 = 2

    I wondered why I had two different results.

    Anyways, take the binary operation a°b = (a+b)/2.
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