In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , , , or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a family of sets over . Example If is the set , then all the subsets of are (also denoted or , the empty set or the null set) and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. An indicator function or a characteristic function of a subset A of a set S with the cardinality |S| = n is a function from S to the two elements set {0, 1}, denoted as IA: S → {0, 1}, and it indicates whether an element of S belongs to A or not; If x in S belongs to A, then IA(x) = 1, and 0 otherwise. Each subset A of S is identified by or equivalent to the indicator function IA, and as the set of all the functions from S to consists of all the indicator functions of all the subsets of S. In other words, is equivalent or bijective to the power set . Since each element in S corresponds to either 0 or 1 under any function in , the number of all the functions in is 2n. Since the number 2 can be defined as (see, for example, von Neumann ordinals), the is also denoted as . Obviously holds. Generally speaking, XY is the set of all functions from Y to X and . Cantor's diagonal argument shows that the power set of a set (whether infinite or