4. Suppose that the sets A, B, and their complements A', B', are recursively enumerable (r.e.). Is the set C = A' union B recursive ? Present a formal proof. 5. Suppose that the set D is only r.e. Is the set E = D union D' recursive ? 6. Show that for any positive integer k, the line segment (1/2k, 1/k) contains uncountably many irrational numbers.
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To prove that the set C = A' union B is recursive, we need to show that there exists a Turing machine that can decide whether a given input belongs to C or not. Since A and B are recursively enumerable, there exist Turing machines TM_A and TM_B that can enumerate Show more…
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