Question

Show That the Collection of Decidable Languages Is Closed Under the Operation of Complementation

A) Turing Machine Proofs
B) Language Complexity Analysis
C) Decidability Closure Proof
D) Complementation in Computability

Answer 1

To show that the collection of decidable languages is closed under the operation of complementation, we need to prove that if a language is decidable, then its complement is also decidable.

Step-by-step explanation:

The collection of decidable languages is closed under the operation of complementation. A language is decidable if there exists a Turing machine that accepts every word in the language and rejects every word not in the language. To show that the collection of decidable languages is closed under complementation, we need to prove that if a language is decidable, then its complement is also decidable.

1. Let L be a decidable language.
2. There exists a Turing machine M that decides L.
3. We can construct a Turing machine M' that decides the complement of L.
4. M' simply simulates M, accepting if M rejects and rejecting if M accepts.
5. Therefore, the complement of L is also decidable.

This proof shows that the collection of decidable languages is closed under the operation of complementation.

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