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Problem 4 (Sets defined inductively) [30 marks] Consider the set S of strings over the alphabet...

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Problem 4 (Sets defined inductively) [30 marks] Consider the setS of strings over the alphabet {a, b} defined inductively asfollows: • Base case: the empty word λ and the word a belong to S •Inductive rule: if ω is a string of S then both ω b and ω b abelong to S as well. 1. Prove that if a string ω belongs to S, thenω does not have two or more consecutive a’s. 2. Prove that for anyn ≥ 0, if ω is a string of length n over the alphabet {a, b} thatdoes not have two or more consecutive a’s, then ω is a string ofS.

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