axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.) Power sets The power set of a set is the set of all subsets of . The empty set and itself are elements of the power set of , because these are both subsets of . For example, the power set of is . The power set of a set is commonly written as or . If has elements, then has elements. For example, has three elements, and its power set has elements, as shown above. If is infinite (whether countable or uncountable), then is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of with the elements of will leave some elements of unpaired. (There is never a bijection from onto .) Partitions A partition of a set S is a set of nonempty subsets of S, such that every element x in S is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S. Basic operations Suppose that a universal set (a set containing all elements being discussed) has been fixed, and that is a subset of . The complement of is the set of all elements (of ) that do not belong to . It may be denoted or . In set-builder notation, . The complement may also be called the absolute complement to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers. Given any two sets and , their union is the set of all things that are members of A or B or both. their intersection is the set of all things that are members of both A and B. If , then and are said to be disjoint. the set difference (also written ) is the set of all things that belong to but not . Especially when is a subset of , it is also called the relative complement of in . their symmetric difference is the set of all things that belong to or but not both. One has . their