*Bounty: 50*

*Bounty: 50*

I have a question I was unable to do, from a last test I had.

This is the question:

Suppose that there is a language $A neq varnothing ,sum{_{}}^{*}$ such that $A in CoNP – CoNPC$.

Determine which of the following statements is correct:

- In this case it is possible to find a language $B$ such that $B in CoNPCcap P$, since it follows that $CoNPCcap P neq varnothing $, and therefore $P=NP$.
- The existence of the language $A in CoNP – CoNPC$ assures us that there is at least one $B in CoNP$ so that $B nleq _p A $, so if we assume in the negative that $B in P$, we can have a contradiction to the non-existence of conversion $B nleq _p A $, because there can always be a conversion from a problem in $P$ to a $CoNP$ problem. That is, $B in CoNP$ and also $B notin P$, and therefore $P neq CoNP$.
- Since there is a language $A in CoNP$ such that $A notin CoNPC$ derives as $CoNP – CoNPC neq varnothing $, therefore any problem $B$ in $CoNP$ can be solved by converting to problem $Cin P$. That is, it follows that $CoNP subseteq P $. The bride in the other direction we have already seen, so $P = CoNP$.
- Nothing can be determined from the data regarding equality or inequality between $P$ and $NP$
- None of the above claims are true.

In the question I am told that:

$A neq varnothing ,sum{_{}}^{*}$ and $A in CoNP – CoNPC$

And according to this you have to choose one of the 5 options.

I understood the question like this:

A, it’s some language, not empty. Which belongs only to CoNP.

The answer I think is correct is 2.

- There is no connection between language B and language A, so it is disqualified.
- True, B can also belong to CoNPC, and there can be no reduction from CoNPC to A.
- Can’t figure out that answer
- It could also be true that they did not talk about it in question.
- Disqualified maybe 2 or 4 are correct

I can not figure out if the answer is 2 or 4.