 So far we have discussed preference aggregation rules, which map profiles of individual preference orderings to social preference relations. We now consider social choice rules, whose output, instead, is one or several winning alternatives. Formally, a social choice rule, \(f\), is a function that assigns to each profile \(\langle R_1, R_2 , \ldots ,R_n\rangle\) (in some domain of admissible profiles) a social choice set \(f(R_1, R_2 , \ldots ,R_n) \subseteq X\). A social choice rule \(f\) can be derived from a preference aggregation rule \(F\), by defining \(f(R_1, R_2 , \ldots ,R_n) = \{x \in X\) : for all \(y \in X\), \(xRy\}\) where \(R = F(R_1, R_2 , \ldots ,R_n)\); the reverse does not generally hold. We call the set of sometimes-chosen alternatives the range of  \(f\).[6]