If you really wanted to ensure this with certainty just use the natural numbers to parameterize an aspect of a general problem. Assume there are N foo problems in the training set, then there is always a case N+1 parameter not in the training set, and you can use this as an indicative case. Go ahead and generate an insane number of these and eventually the probability that the Mth instance is not in the set is effectively 1.

Edit: Of course, it would not be perfect certainty, but it is probabilistically effectively certain. The number of problem instances in the set is necessarily finite, so if you go large enough you get what you need. Sure, you wouldn't be able to say there is a specific problem instance not in the set, but the aggregate results would evidence whether or no the LLm deals with all cases or (on assumption) just known ones.