m25-hw1-solutions

m25-hw1-solutions - Math 25: Advanced Calculus Fall 2010...

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Math 25: Advanced Calculus Fall 2010 Homework # 1 – Solutions Due Date: Friday, October 1 A.5.2: Show that there are infinitely many prime numbers. Proof: We will use proof by contradiction. Suppose to the contrary that there are only finitely many primes, and call them p 1 ,p 1 ,...,p r . Consider the number N = p 1 · p 2 · ... · p r + 1 . Since N ÷ p i has remainder 1 for all of our primes p i , we see that N is not divisible by any prime p i . We know, however, that any natural number can be written as a product of prime numbers. This contradicts our assumption that p 1 ,...,p r are the only prime numbers. ± A.6.1: Prove the following assertion by contraposition: If x is irrational, then x + r is irrational for all rational numbers r . Proof: We want to show that P Q , where P : x is irrational Q : x + r is irrational for all rational numbers r . In order to employ proof by contraposition, we need to show that Q ⇒∼ P , and we begin by negating the statements P and Q : Q
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m25-hw1-solutions - Math 25: Advanced Calculus Fall 2010...

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