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What is the meanning of this propisition ?

  1. Oct 22, 2012 #1
    hi , this proposition is from artin's book ,

    it says :
    Proposition : suppose an associative law of composition is given on a set S.
    There is a unique way to define , for every integer n , a product of n elements
    a1, …… , an of S ( we denote it by [a1 …an ] ) with the following properties :
    -The product [a1] of one element is the element itself
    -The product [a1 a2] of two elements is given by the law of composition ;
    -For any integer I between 1 and n , [a1 .. an]=[a1….ai][a(i+1)…..an]

    and I didn't understand excatly the meaning which the proposition says , so can anyone make it clear to me please ? and if there an example it'll be more better :)

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    Last edited: Oct 22, 2012
  2. jcsd
  3. Oct 22, 2012 #2


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    Here's a simple example, using addition of real numbers.

    You only define what addition means for adding two numbers (not more than two).

    The associative law says that (x + y) + z = x + (y + z) for any three numbers x, y, and z.

    The proposition says that if the associative law is true, there is a unique "answer" for the sum of ANY number of integers, a1 + a2 + a3 + ... + an, and the answer does not depend how you split the sum up into pairs of numbers using ()'s.
  4. Oct 23, 2012 #3
    thank you very much :)
    I have understood :)
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