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Mathematics
Linear and Abstract Algebra
How can we use induction to prove the generalized associative law?
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[QUOTE="fresh_42, post: 6016667, member: 572553"] So this is the induction hypothesis: Suppose that [B][if][/B] [B]for any[/B] ##k < n## that [B]any [given] bracketing[/B] of the product of ##k## elements [B]then ...[/B] Now we have to prove: Suppose that [B]if [/B]##k=n## that [B]some[/B] [B]given bracketing[/B] of the product [B]then[/B]: [B]... [/B] It is part of our IF clause. We have some given bracketing of ##n## elements. In case it is of the form ##a_1(a_2(\ldots a_n)\ldots )## we are done. All other cases are of the form ##(a_1 \ldots a_k)(a_{k+1}\ldots a_n)## with some bracketing of the two parts and ##1<k##. I assume we have commutativity, too, so we may assume ##k<n-1##, too. For the non commutative case, the expression ##a_1\ldots a_n## has to be explained, as e.g. in the case of functions. It is sloppy to write ##fg## and not mentioning whether ##g(f(x))## or ##f(g(x))## is meant. It's usually the latter, but purists prefer the first version. [/QUOTE]
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Mathematics
Linear and Abstract Algebra
How can we use induction to prove the generalized associative law?
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