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Unformatted text preview: 4. Let Σ denote a finite set of symbols, i.e., Σ denote an alphabet. Let W ⊂ Σ *, be a set of strings over Σ . Answer the following two parts for this question: (a) Prove W * ⋅ W * = W *, where the notation “ ⋅ ” means string concatenation. ( Hint: There are two parts to this problem: (1) prove W * ⋅ W * ⊂ W *; and (2) prove W * ⊂ W * ⋅ W *. You may find these rules useful: x ∈ W * ⇔ x ∈ W m for some m ≥ 0 (by the definition of W *); λ ∈ W * (by the definition of W *); if A ⊂ B , then AC ⊂ BC ; and λ⋅ A = A , where A , B , C denote sets of strings.) (b) Prove using induction on n ≥ 1 that ( W *) n = W * ⋅ W * ⋅ ⋅ ⋅ W * (i.e., W * concatenated with itself for n times) = W *....
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This document was uploaded on 07/14/2011.
 Spring '09

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