Unformatted text preview: nx. 3. Prove by induction that for every n ∈ N the number 7 n4 n is divisible by 3. 4. Find and explain the ﬂaw in the proof by induction that was presented in class of the claim that all giraﬀes are the same height (see also problem A.8.9 in the textbook). 5. Answer problems A.9.1 and A.9.2 in the textbook. 6. Prove that the set N × N = { all pairs ( m,n ) where m,n ∈ N } is countable. That is, show that one can write down a list of pairs ( m 1 ,n 1 ) , ( m 2 ,n 2 ) , ( m 3 ,n 3 ) ,... covering all the elements of N × N (the list may contain repetitions but any pair ( m,n ) of natural numbers must appear at least once). 2 1 Last week’s HW guidelines also still apply — refresh your memory by rereading them. 2 You do not need to write a precise formula for the general term ( m k ,n k ) of this list, as long as you provide suﬃcient detail about how to construct the list that convinces the reader that you understand what you are talking about....
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 Winter '11
 Wiley
 Differential Equations, Calculus, Equations

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