Set theory

Set theory, branch of mathematics or symbolic (mathematical) logic in which systems are analyzed by membership in and exclusion from sets. A set is any specified collection of elements or members. In the notation of set theory, members of a set are enclosed in braces as, {a,b,d,x}. Alternatively, a rule defining inclusion in the set may appear in braces. The expression {xx is a U.S. state bordered by 3 or more other states} defines the set of all U.S. states bordered by 3 or more other states; this is a finite set, having a definite number of members. The set of all positive even numbers is an example of an infinite set, one with an infinite number of members. It might be represented as {2,4,6,8,…} or as {xx/2 is an integer greater than 0}. Null sets, those with no members (e.g., all U.S. presidents who are female, nonwhite, or non-Christian) are represented as { }. Sets that have exactly the same members (the set of all U.S. presidents who have served more than 2 terms, the set of all presidents crippled by polio) are said to be equal sets, while those with the same number of members but not necessarily the identical membership are equivalent. Set A is a subset of set B if all of the members of set A are contained in set B. (The set of all equal sets is thus a subset of the set of all equivalent sets, but not vice versa). The union of 2 sets A and B (A B) is the combined membership of those 2 sets. The intersection of sets AandB (An ∩ ) is the set of those elements belonging to both set A and set B. Disjoint sets have no common members and thus their intersection is a null or empty set (the set of all women, the set of all U.S. presidents). If A is a subset of B, the complement of A consists of all members of B not in A. Set theory, which owes much to the techniques of symbolic logic established by 19th-century English mathematician George Boole, was developed by 19th-century German mathematician Georg Cantor. It is of particular importance in representing logical relationships and approaches to algebraic solutions.

See also: Mathematics.