Abstract: Function one direction singular rough, dual of function one direction singular rough sets and function two direction singular rough sets were given. They arc three types of forms about function trough sets. They came from improved S-rough sets by introducing function concept to S-rough sets. Function rough sets came from improved Z. Pawlak rough sets by introducing function concept to Z. Pawlak rough sets. This paper gave the relation between function Srough sets and function trough sets, the relation between function trough sets and Z. Pawlak rough sets, the relation between function S-rough sets and function rough sets. Using these results, this paper presented the generation of the interval discretization for function and finite element set, the principle of function discretization and element generation,the information law which was generated by function S-rough sets, the principle of the dynamic properties for function equivalence class and supplementary and deletion for properties, the separation-composition principle for data, the attribute characteristics of the dynamic separation-composition for information law, the invariance principle of the dynamic separation-composition for information law and so on. Using these concepts and results, the application of the separalion-composition for information law, the application of embedment separation for information images and the separalion-identification of embedded information images were given. In short, function S-rough sets and S-rough sets are a new research direction in rough sets.

Key words: Function S-rough sets, Function rough sets, Relation theorem, Information law generation, Separation-Composition principle, Information image embedment, Application