# set theory – Notions of “completeness” and “sufficiency” of a mathematical model

I’m modelling a real-world problem as having instances $$i$$ in a set $$P$$. The structure of the problem and the model itself are irrelevant to my question so I’ll omit them.

I define certain restrictions $$A$$ on $$P$$ using logical formulae over structure of $$i$$.
$$A$$ is a necessary condition, i.e. any problem in the problem domain, if represented using $$i in P$$ would satisfy $$A$$. We can say that $$P$$ under $$A$$, i.e. $$P’ = {p : p in P land p~text{satisfies}~A}$$.

But it’s possible that $$exists i’ in P’$$ such that $$i’$$ is a valid mathematical structure but doesn’t actually correspond to valid any real-world problem.

What is the standard terminology to say

1. All valid problem instances are a part of $$P’$$ (I’m informally calling this “sufficiency”)
2. All problem instances which are a part of $$P’$$ are valid (I’m informally calling this “completeness”)

I may then use this terminology to say that in my example, $$P’$$ is sufficient but incomplete (replacing these two words with the actual terminology)