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An essential ingredient in many examples of the conflict between quantumtheory and noncontextual hidden variables (e.g., the proof of theKochen-Specker theorem and Hardy's proof of Bell's theorem) is a set of atomicpropositions about the outcomes of ideal measurements such that, when outcomenoncontextuality is assumed, if proposition $A$ is true, then, due toexclusiveness and completeness, a nonexclusive proposition $B$ ($C$) must befalse (true). We call such a set a {\em true-implies-false set} (TIFS) [{\emtrue-implies-true set} (TITS)]. Here we identify all the minimal TIFSs andTITSs in every dimension $d \ge 3$, i.e., the sets of each type having thesmallest number of propositions. These sets are important because each of themleads to a proof of impossibility of noncontextual hidden variables andcorresponds to a simple situation with quantum vs classical advantage.Moreover, the methods developed to identify them may be helpful to solve someopen problems regarding minimal Kochen-Specker sets.

Minimal true-implies-false and true-implies-true sets of propositions in noncontextual hidden variable theories