$P$ means polynomial complexity.
$S\_p$ is the class of all $P$\_random set, and $S\_{pc}$ is the class of all $P$ noncomputable sets, is $S\_p \bigcap S\_{pc}$ empty? If not empty, any example?


what is the result, if we replace $P$ complexity with $NP$?


Moreover, $S$ is the class of all random sets, and $S\_c$ is the class of all incomputable sets, $S\_{c}\bigcap S $ is not empty, what is a set in $S\_{c}\bigcap S $,an immune set, productive set, or set of any other kind.


Is the difference class empty?


$P$ means polynomial complexity.
$S\_p$ is class of all $P$\_random sets, and $S\_{pc}$ is class of all $P$ incomputable sets, is $S\_{pc} \setminus S\_p$ empty? If not empty, any example?


what is the result, if we replace $P$ complexity with $NP$?


Moreover, $S$ is class of all random sets, and $S\_c$ is class of all incomputable sets, $S\_{c}\setminus S $ is not empty, what is a set in the class $S\_{c}\setminus S $,an immune set, productive set, or set of any other kind.


I have restored the original post with clarification, and ask the question I intend in the final part.



Relationship between P-noncomputable and P-random sets
No, the intersection is not empty. 


To see this, notice that we can easily find sets $A\subset\mathbb{N}$ that are random with respect to much stronger notions of complexity. And indeed, there will be uncountably many such sets $A$. Not all these can be computable. So we find $A$ that are both random and undecidable, as desired. 


Regarding the update, yes, some noncomputable sets are not random. For example, take any noncomputable sequence, and repeat every bit twice. Now it is still not computable, but also not random.


