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Write definitions for the following:

  1. ${ExhaustivePartDecomposition}$

  2. ${PartPartition}$

  3. ${PartwiseDisjoint}$

These should be analogous to the definitions for ${ExhaustiveDecomposition}$, ${Partition}$, and ${Disjoint}$. Is it the case that ${PartPartition}(s,{BunchOf}(s))$? If so, prove it; if not, give a counterexample and define sufficient conditions under which it does hold.