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74hwsol4 - Solutions to Homework Assignment 4 Math 74 Fall...

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Solutions to Homework Assignment 4 Math 74, Fall 2006 October 6, 2006 1. Definition 1. A natural number n is a perfect square if there exists an integer k such that n = k 2 . Theorem 2. If n N then n Q if and only if n is a perfect square. Proof. Clearly if n N is a perfect square, then n N Q . Suppose, on the other hand, that n N is not a perfect square. Define B = k N | k n Z . We will show that B is nonempty. Proceeding by way of contradiction, suppose that B is not empty. By the Well-Ordering Principle, B has a least element. Let b = min B . Then b n Z since b B . Now choose an integer m such that m < n m + 1 . By the Well-Ordering Principle, it is obvious that such an integer m exists, and that it is unique. Also, since n is not a perfect square, n = m + 1, and so n < m + 1. Now define k = b ( n - m ) . Notice that k is an integer, since k = b n - bm is the difference of two integers. Also, our choice of b guarantees that 0 < n - m < 1. So we conclude that 0 < k < b. Since k n = bn - m ( b n ) is the difference of two integers, and is therefore an integer, we deduce that k B . But k < b contradicts the fact that k = min B .
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