All P problems are in NP. If you can solve a problem in polynomial time, you could simply "check" the problem on an input by solving the problem and seeing if t

All P problems are in NP. If you can solve a problem in polynomial time, you could simply "check" the problem on an input by solving the problem and seeing if the test input were in the set of solutions. It will clearly take P time to perform that "check" (because you can solve the problem in P time). This means all problem in P must be in NP. The error is that not all problems in NP are "extremely difficult for computers to solve". The difficulty of solving problems in NP varies drastically between particular problems.