I seem to be having trouble understanding the connection between the formal definition of co-NP and how problems are concluded to be in it. co-NP is defined to be the class containing the languages that are complements to languages in NP (I'm using this definition).
So, given this problem: "given a finite set of integers, is there a non-empty subset that sums to zero?" is in NP. How do i conclude that this problem: "given a finite set of integers, does every non-empty subset have a non-zero sum?" is in the class co-NP?
In other words, how do i know that the second problems corresponding language L2 is the complement to the first problems corresponding language L1?
Perhaps i'm missing something fundamental?!
Edit: It seems I was not clear enough so hear is a clarification: My question was why does the complement language L2 represent the second problem if it contains many seemingly random strings?
In other words, how are these seemingly random strings instances of the second problem?