Just stepping into complexity theory, I am befuddled by this notion of a certificate and can't find any utility of this concept.
From my understanding, a certificate is used when you are trying to ascertain whether a problem is NP...a problem is NP if you can verify a p-time solution exist or not (co-NP). I cannot see why you would need a certificate to perform this solution finding. And how is certificate different from "verification".
For example, if I were to try to prove whether all elements in a set is divisible by some number (say 3). Say I then brute force divide all elements by 3 and sees that indeed all elements in this set is divisible by 3. Now where in this process would a certificate come into play?
Also, how would you go about to physically construct a certificate?