hw6 - b c in which the total number of b ’s and c ’s is...

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COT3100 Summer’2001 Assignment #6 (Optional). Assigned: 07/17, due: 07/26 before Final. Indicate your name, SSN and your recitation session number. 1. Let X = { aa , bb } and Y = { λ , b , ab } be two languages over { a , b }. a) List the strings in set X Y . b) List the strings of the set Y * of length three or less. c) How many strings of length 6 are there in X * ? 2. Let A , B and C be sets of strings. Prove or disprove that if A B , then A C B C . 3. Give the regular expressions for the following languages: a) The set of strings of length two or more over the alphabet { a , b } in which any occurrence of b is immediately preceded by a . b) The set of strings over { a , b } that do not contain a substring ab . c) The set of strings over { a , b }, in which the substring aa occurs exactly once. d) The set of strings over { a ,
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Unformatted text preview: b , c } in which the total number of b ’s and c ’s is three. 4. Let L be the language over { a , b } generated by the recursive definition: i) Basis: λ∈ L ii) Recursive step: If u ∈ L then aaub ∈ L iii) Closure: A string w is in L only if it can be obtained from the Basis by finite number applications of the Recursive step. Prove by induction that for any u ∈ L the number of a ’s in u is twice the number of b ’s. 5. Prove or disprove that if X * = Y * , then X = Y. 6 Write down the transition function for the following DFA: 7. Construct the DFA to recognize the language of all strings over { a , b } in which the number of a ’s is divisible by 3. Give the diagram and transition table. b b b b a a a a 1 2 3...
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