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# hmk5_key - COT 3100 Spring 2001 Homework#5 Assigned April 9...

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COT 3100 Spring 2001 Homework #5 Assigned: April 9, 10. Due: April 18, 19 in recitation. Problem #1 Let A , B and C be sets of strings. Prove or disprove that A ( B - C ) = A B - A C . Only one subset relation can be proved, namely we can prove that A B - A C A ( B - C ). It actually means that A B - A C is “smaller” then A ( B - C ), because there might be a string that belongs to both A B and A C , although it can not be represented as some prefix from A and a suffix that belong to both B and C. So, here is a counterexample that disproves A ( B - C ) A B - A C . A ={ a , ab }, B ={ b }, C ={ λ }. B - C ={ b }, A ( B - C )={ ab , abb }, A B ={ ab , abb }, A C ={ a , ab }, and A B - A C = { abb }. So, ab A ( B - C ), but ab A B - A C , so it’s not the case, that always A ( B - C ) A B - A C . Problem #2 Give regular expressions to describe the set of strings in (a b) * that a) contain exactly one occurrence of a ; b * ab * b) contain exactly two occurrences of a ; b * ab * ab * c) begin with b ; b ( a + b ) * d) end in aba ; ( a + b ) * aba

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