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PS: There are no functions from the empty set to any set. If we denote the empty set as E and any other set as S, then the set of functions from E -> S is equivalent to the set { subsets K of ExS : if (a,b) appears in K, then no other element of K is (a,c) for any c}. In other words, the set of relations between E and S in which no element of E is related to more than one element of S. But ExS = E, and the set of subsets of E is E, and #(E) = 0, so obviously the number of subsets of ExS satisfying the necessary property is also 0.