A decision problem is a problem with a yes or no answer. An NP problem is one where a "yes" answer can be verified quickly if they also give you some proof or "

A decision problem is a problem with a yes or no answer. An NP problem is one where a "yes" answer can be verified quickly if they also give you some proof or "witness" of it. If I say "is there a route that goes through all these cities that's less than 500 miles long", and you answer yes, you can prove that such a route exists by showing me one of these and I can simply check how long the route is. So the problem is in NP because a "yes" answer can be proven or checked quickly. However, if you say "no, there is no such route", there's not obviously any way to quickly show that. Despite that, the problem is still in NP because to be in NP there only needs to be a quickly-checkable proof of a "yes" answer. If you want a quickly checkable proof of a "no" answer, you need a separate class of problem called co-np. A problem can also be in np and co-np at the same time, if both "yes" answers and "no" answers can have a proof that can be checked quickly.