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If � 1 L 1 ​ is decidable and � 2 L 2 ​ is recognisable then (select all that apply): The following checkbox options contain math elements, so you may need to read them in your screen reader's “reading” or “browse” mode instead of “forms” or “focus” mode. Choice 1 of 3: � 1 ∪ � 2 L 1 ​ ∪L 2 ​ is decidable. Choice 2 of 3: � 1 ∪ � 2 L 1 ​ ∪L 2 ​ is not decidable. Choice 3 of 3:Not enough information to answer.

Question

If � 1 L 1 ​ is decidable and � 2 L 2 ​ is recognisable then (select all that apply):

The following checkbox options contain math elements, so you may need to read them in your screen reader's “reading” or “browse” mode instead of “forms” or “focus” mode. Choice 1 of 3: � 1 ∪ � 2 L 1 ​ ∪L 2 ​ is decidable. Choice 2 of 3: � 1 ∪ � 2 L 1 ​ ∪L 2 ​ is not decidable. Choice 3 of 3:Not enough information to answer.

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Solution

Choice 1 of 3: L1 ∪ L2 is decidable.

This is because if a language is decidable, then it remains decidable even when unioned with a recognizable language. The decidable language provides a method for always determining membership, and this method can be used for the unioned language.

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