ass4-soln

# ass4-soln - Math 239 Assignment 4 This assignment will not...

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This assignment will not be marked. Do not hand in. Math 239 Assignment 4 1. For each of the following sets A , determine whether A * is uniquely created. Prove your assertion in each case. (Hint: to prove a set A * is uniquely created, try using induction on the length.) (a) A = { 00 , 010 , 0111 } (b) A = { 010 , 0010 , 01001 } . Solution: (a) In this case, A is uniquely created. We prove by induction on n that every σ A of length n is uniquely created. If n = 0 then the only string in A * of length 0 is the empty string, which is uniquely created. Suppose n 1 and that the statement holds for smaller values of n . Let σ = a 1 ...a n . So a 1 = 0 and n 2. Consider a 2 . If a 2 = 0 then σ = 00 a 3 n where a 3 n A * and by the induction hypothesis a 3 n is uniquely created. Therefore σ is uniquely created. If a 2 = 1 then σ = 01 a 3 n and n 3. Consider a 3 : * If a 3 = 0 then σ = 010 a 4 n where a 4 n A * and by the induc- tion hypothesis a 4 n is uniquely created. Therefore σ is uniquely created. * If a 3 = 1 then σ = 0111 a 5 n where a 5 n A * and by the induc- tion hypothesis a 5 n is uniquely created. Therefore σ is uniquely created.

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ass4-soln - Math 239 Assignment 4 This assignment will not...

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