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September 16th, 2009, 09:27 AM   #1
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addition with parenthesis revisited..

Hello, I posted a question about a problem in the Spivak book a while ago, I just noticed that there was a part c to the question that is kind of confusing.. I'd like some help please

a + b + c denotes a+(b+c) , a + b + c + d denotes a+(b+(c+d))..a1 + ...... + an denotes a1 + ( a2 + ( a3 .....+ (an-1 + an)))

Now the question asks me to show that S(a1, ...... , ak) = a1 + ..... +ak

the hint says that there much exist a S1(a1, ...... , al) + S2(al+1, ...... , ak) = S(a1,..... , ak).

My first instinct was to somehow use the closure under addition property with the elements of the set S.. but I really don't understand the hint - the biggest thing that concerns me is that the addition operation between these two sets has not been specifically defined? how do they expect me to add the sets when the operation is not explicitly defined?

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September 17th, 2009, 06:00 AM   #2
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You can't prove anything about S unless it was already defined. It wouldn't have been defined as a set.
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