  A new notion of partition-determined functions is introduced, and several
basic inequalities are developed for the entropy of such functions of
independent random variables, as well as for cardinalities of compound sets
obtained using these functions. Here a compound set means a set obtained by
varying each argument of a function of several variables over a set associated
with that argument, where all the sets are subsets of an appropriate algebraic
structure so that the function is well defined. On the one hand, the entropy
inequalities developed for partition-determined functions imply entropic
analogues of general inequalities of Pl\"unnecke-Ruzsa type. On the other hand,
the cardinality inequalities developed for compound sets imply several
inequalities for sumsets, including for instance a generalization of
inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide
partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian
groups. All proofs are elementary and rely on properly developing certain
information-theoretic inequalities.
