A new notion of partition-determined functions is introduced, and several basic new 
inequalities are presented for the entropy of such functions of independent random vari- 
ables, 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 sev- 
eral 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 deŽned. The method of 
proof of the compound set cardinality inequalities relies on the entropy inequalities. The 
compound set inequalities imply, in turn, several inequalities for sumsets, providing for 
instance partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian groups.