Unformatted text preview: Hint: ²ollow the proo± that √ 2 is not a rational number in the book. 5. Section 1.2. Problem 1. Use induction on n to show that  u n  = n  u  ±or all srings u and integers n ≥ 0. Hint: Use the ±act that u i +1 = u i u and u = λ . 6. Section 1.2. Problem 8. Show that ( L 1 L 2 ) R = L R 2 L R 1 . Hint: Use the ±ollowing ±act (problem 2) as given. ²or any strings u,v over some alphabet ( uv ) R = v R u R . Note that u R is the reverse o± string u . ( L ) R is the set o± reverses o± strings in L . 7. Section 1.2. Problem 9. Show that ( L * ) * = L * ±or all languages L . 8. Section 1.2. Problem 11(b). ²ind a grammar ±or ∑ = { a,b } that generates the sets o± all strings with at least one a . 9. Section 1.2. Problem 14(b). Give a grammar to generate the language L 2 = { a n b 2 n : n ≥ } . 10. Give a grammar to generate the language L * 2 , where L 2 is deﬁned in the previous problem. 1...
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 Fall '08
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 Lefthandedness, finite state automata

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