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PROBLEM 1. Show that the Hamming distance is a metric on V" the set of all words of length n. PROBLEM 2. Compare the Hamming metric and the Levenshtein metric on the set V". Is either metric, in general, always less than or equal to the other? Prove your assertion. PROBLEM 3. We say that a code of length p corrects n errors if every word in VP is within distance n of a unique codeword in the code. (a) In VS. find an example of a code C that has four codewords and corrects two errors. (b) Show that four codewords is the maximal size for a code in V8 that corrects two errors. PROBLEM 4. For each of the following, find the Levenshtein distance between the two se- quences, x andy. In cases where the sequences have equal length, also compute the Hamming distance. (a) x = ACGGTAT and y = GGTAG (b) x = CTGGTAC and y = CTAGATC

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