Relation between NP, recursive and recusive enumerable

Question

I) Every language in NP is recursive.

II)Every language in NP is recursively enumerable.

Which of the statements is /are true?

A. I only

B. II only

C. Both I and II

D Neither I nor II

Answer 1

p⊆Np⊆recursive⊆recursive enumerable

just remember this

Comments

NP is proper subset of recursive which is in turn a proper subset of REL. src: https://cs.stackexchange.com/questions/90659/what-is-the-relation-between-np-np-hard-problems-and-recursive-r-e-languages-an

Answer 2

i think every problem in NP is decidable means it is recursive .and every recursive problem is recursively enumerable so we can say that both are true,correct me if i am wrong ..

Comments

yes you are right..!!

answer is C

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@Sidhi: Sir,

I have a doubt what does "every problem in NP is decidable " means?

What we are deciding for given NP problem?

Are we deciding proposed solution is correct or not?

Or are we deciding whether any solution exists for given problem?

Sorry, if I asked a lame question :P

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i can't understand what you want to ask exactly..may below link will help you to understand, check it out.

https://www.quora.com/Theoretical-Computer-Science-Are-all-P-languages-decidable-Are-all-NP-languages-decidable

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@Sidhi: thanks for that wonderful link.. Concept is pretty clear now!!

• NP is the class of languages that can be recognized by a non deterministic Turing machine
• A decidable language is a formal language for which there exists a Turing machine which will, when presented with any finite input string , halt and accept if the string is in the language, and halt and reject otherwise. 
• Since any language in NP can be recognized by a non deterministic TM,  there is a TM that always halts when presented with any finite input string of the language. 
• Hence NP is decidable
•  Since $P\subset NP$, any language in P is also decidable.

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p⊆Np⊆recursive