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If $S$ is a set closed under an associative operation, prove that no matter how you bracket $a_1 \cdot a_2 \cdot a_3 \cdots a_n$, retaining the order of the elements, you get the same element in $S$.

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    I hope that you don't mind that I changed the title to more accurately reflect the question.2012-09-20

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You can try proving it by induction on n (i.e. on the number of elements operated on).

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    Yes ,Induction will definitely do...but may anyone please frame proposition on which induction is to be applied?2012-09-21