uestion 1. Explain what is wrong with the following proofs: - (4 points) Theorem: Every...
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uestion 1. Explain what is wrong with the following proofs: - (4 points) Theorem: Every cven integer greater than two is the sum of two primes. Proof. Let's call the integer in question n. We prove this theorem by induction on even integers n. The base case is n=4 as 4 is the first even integer greater than two. Base case n=4:4=2+2, so the thcorem holds in this casc. Inductive step: Let nbc an cven integer greater than two. Assume that n is the sum of two primes p and q. Then n+2 is the sum of the two primes p+2 and q, so the next cven integer n+2 is also a sum of two primes. It follows by induction that cvery cven integer greater than two is the sum of two primes. - (4 points) Theorem: Let be an arbitrary positive real number. Then, for all integers n1 it holds that n1=1. Proof: By induction on n. For n=1, we have n1=0=1, so the base casc holds. Inductive step: Suppose that the theorem is true for n=k. That is, k1=1. For n=k+1, wc have (k+1)1=k=(k1)k1k=1k1k=1 which completes the inductive step
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