20111004s - on p. 131, it implies that ℒ g is recursive....

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Theory of Computation Solutions to Homework 1 Problem 1. Two disjoint languages g and G are called recursively separable if there exists a recursive language such that g ∩ ℛ = ∅ and G ⊆ ℛ . Suppose g and G are recursively separable languages. Show that if both g and u g ∪ ℒ G are recursively enumerable, then g is recursive. Proof. Without loss of generality, assume g and G are recursively separable with G ⊆ ℛ . Obviously, G ⊆ ℒ u g . Let TM accept u g ∪ ℒ G . Because u g ∪ ℒ G = ℒ u g , u g is also accepted by , thus recursively enumerable. Using the idea in Lemma 10
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Unformatted text preview: on p. 131, it implies that ℒ g is recursive. Problem 2. Prove that the subsets of distinct primes form an uncountable set. Proof. Denote the ± th prime as ² ³ . It is easy to show that there is a bijection between ℕ and primes ´ ∶ µ1,2,3,4,. ..,±,. .. ¶ U µ2,3,5,7,. .., ² ³ ,... ¶ . Therefore, primes are countable. Thus the problem is equivalent to asking whether a function · exists such that · ∶ ℕ U 2 ℕ is a bijection. Cantor’s theory says no such · exists. Hence the subsets of distinct primes do not form a countable set ....
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