Homework 2

# Homework 2 - MATH 74 HOMEWORK 2(DUE WEDNESDAY SEPTEMBER 12...

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MATH 74 HOMEWORK 2 (DUE WEDNESDAY SEPTEMBER 12) 1. We have seen that if you assume that (1) x + ( y + z ) = ( x + y ) + z for all natural numbers x, y, z then you can prove that various ways of associating the sum of four or five natural numbers must give the same result (see the lecture notes for August 31, and last week’s homework). Most (if not all) of us have no trouble believing that for any n , repeated applica- tions of (1) can prove that any way of parenthesizing a sum of n natural numbers gives the same result as any other. (This is what textbooks implicitly assume when- ever they state something like (1) for three-element sums, and then say “because of this, grouping doesn’t matter.”) But what would a proof of this actually look like? Perhaps more pressing: what would a proper statement of this proposition look like? (What is a “way of parenthesizing”?) Without getting into formalities, we can certainly list all ways one might evaluate a three-term sum without changing

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