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131aw12hw1

131aw12hw1 - Math 131A Problem Set#1 Due January 20...

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Math 131A Problem Set #1 Due January 20 Exercises 1.8, 1.11, 2.4, 3.5, 3.8, 4.1-4.4(a)(b)(e)(k)(v), 4.11, 4.12, 4.14 from the textbook Additional Problem 1: (More Practice with Induction) Prove, by induction, that if p ( x ) is a polynomial of degree n 1 with coefficients from R , then p ( x ) has no more than n roots in R . (Here, we are counting roots with multiplicities . For example, the polynomial p ( x ) = x 2 ( x - 1)( x 2 + 1) has three roots in R , namely 0, 0, and 1.) There is another induction principle, called the principle of Strong Induction , which states the following: Suppose that P ( n ) is a statement such that: P ( m ) is true; Whenever P ( k ) is true for all m k < n , then P ( n ) is true. Then P ( n ) is true for all n m . The main difference between ordinary induction and strong induction is that strong induction allows you to assume the truth of P for all earlier values, not just the previous value. Despite this seemingly major difference, these principles are actually equivalent.

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131aw12hw1 - Math 131A Problem Set#1 Due January 20...

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