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