Hw5sp00 - 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

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COT3100-01, Spring 2000 Assigned: 4/12/2000 S. Lang Assignment #5 (40 pts.) Due: 4/19 in class 1. Use induction on n 0 to prove the following summation identity: . 0 2 ) 2 ( ) 1 ( 2 ) 1 ( = + - = - n i n n n i i 2. Suppose a function f ( n ) is defined for integer n 0 recursively by the following recurrence: f (0) = 7; f (1) = 14; and f ( n ) = f ( n – 1) + 12 f ( n – 2) for n 2. Prove that f ( n ) = 5(4) n + 2(–3) n for all n 0. ( Hint: Use strong induction on n 0, in which the Basis step verifies the formula of f ( n ) for both n = 0 and n = 1.) 3. Answer the following two parts for this question: (a) For any real numbers a and b , prove a 2 + b 2 2 ab . ( Hint: Show that a 2 + b 2 – 2 ab is a square of a real number.) (b) Use induction on n 1 to prove the following inequality: . ) 1 )( ( 2 1 2 1 2 n i i n i n i = = ( Hint: The result k 2 + (1 / k 2 ) 2 which follows from Part (a) by letting a = k , b = (1 / k ), can be useful in the induction step of this part.)
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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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