class of language

Question

1.  L = {<M>|M is a TM and L(M) is countable}
2. L = {<M>|M is a TM and L(M) is uncountable}
what is the class of 1 and 2 recursive/RE/NOT RE

Answer 1

1 is recursive because it is trivial property about language of turing machine as we know langages on finite no of alphabets are always countable and this includes all recursively enumerable languages.

Since 1 is recursive so must be 2 as it is the complement of 1 and recursive languages are closed under complement operation meaning complement of any recursive langauge must be recursive.

Comments

in my opinion both shud be recursive... bcoz laguage accepted by any turing machine is subset of sigma*.. sigma* is countable and subset of sigma* is also countable. we can include all turing machines in first language i think.
as 2nd one is complment of first one, 2nd one also becomes recursive as recursive languages are closed under complmentation.
please confirm this anybody!
@arjun sir

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Both are Recursive.

1. Is the language of turing machine.

2. with finite aphabet and finite length string three is no uncountable lamguages so this set is empty.

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Yes, all above comments are correct :)

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thnx 2 all.........