cartesian product is the set of all ordered pairs such that is an element of and is an element of . Examples: }. }. }. }. }. The operations above satisfy many identities. For example, one of De Morgan's laws states that (that is, the elements outside the union of and are the elements that are outside and outside ). The cardinality of is the product of the cardinalities of and . (This is an elementary fact when and are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true.) The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. Applications Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations. One of the main applications of naive set theory is in the construction of relations. A relation from a domain to a codomain is a subset of the Cartesian product . For example, considering the set of shapes in the game of the same name, the relation "beats" from to is the set ; thus beats in the game if the pair is a member of . Another example is the set of all pairs , where is real. This relation is a subset of , because the set of all squares is subset of the set of all real numbers. Since for every in , one and only one pair is found in , it is called a function. In functional notation, this relation can be written as . Principle of inclusion and exclusion The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically as A more general form of the principle gives the cardinality of any finite union of finite sets: See also Algebra of sets Alternative set theory Category of sets Class (set theory) Dense set Family of sets Fuzzy set Internal set Mereology Multiset Principia Mathematica Rough set Notes References External links Cantor's "Beiträge zur Begründung der transfiniten Mengenlehre" (in German) Concepts in logic Mathematical objects