When 1 to x is sufficient and 1 to x-1 is not, x is required

Title needs improvement

For a predicate f that takes a set as an argument

• $f(S)=0,f(S+\{x\})=1$ Then the thought pattern that x is required to be 1

What are the conditions for being able to say this?

• $f(S)=1\Rightarrow \forall T\supset S,f(T)=1$

• Find one subset that satisfies the condition sometimes $2^N$ instead of doing a search, N is better

• 2^N search is impossible.

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