Complement of NP-complete can be in NP

Question

Here are a couple of questions I struggle with. (We use $A'$ to denote the complement of the problem $A$.)

1. A problem $A$ is NP-complete if and only if $A'$ is in NP?
2. A class problem $A$ is NP-Complete iff $A'$ is in NP? (Is there any difference between these two?)

I got this claim to be true that

If there is an NP-complete language $L$ whose complement is in NP, then the complement of any language in NP is in NP.

And this claim also to be true:

$A$ is NP-hard iff $A'$ is coNP−hard

Then is the upper claim also true?

Answer 1

A problem $A$ is NP-complete if and only if $A'$ is in NP?

This is false. Take $A'=\emptyset$: then we have $A'\in\mathrm{NP}$ but $A=\Sigma^*$ is not $\mathrm{NP}$-complete.

If there is an NP-complete language $L$ whose complement is in NP, then the complement of any language in NP is in NP.

This follows from the following three facts, which you should attempt to prove yourself.

1. The complement of an $\text{NP}$-complete problem is $\text{co-NP}$-complete;
2. if any $\text{co-NP}$-complete problem is in a class $\mathcal{C}$ that is closed under polynomial-time reductions, then every $\text{co-NP}\subseteq\mathcal{C}$;
3. $\text{NP}$ is closed under polynomial-time reductions.