set theory: Definition and Much More from Answers.com

A mathematical term referring to the study of collections or sets. Consider a collection of objects (such as points, dishes, equations, chemicals, numbers, or curves). This set may be denoted by some symbol, such as X. It is useful to know properties that the set X has, irrespective of what the elements of X are. The cardinality of X is such a property.

Two sets A and B are said to have the same cardinal written C(A) = C(B), provided there is a one-to-one correspondence between the elements of A and the elements of B. For finite sets this notion coincides with the phrase “A has the same number of elements as B.” However, for infinite sets the above definition yields some interesting consequences. For example, let A denote the set of integers and B the set of odd integers. The function f(n) = 2n−1 shows that C(A) = C(B). Hence, an infinite set may have the same cardinal as a part or subset of itself.

A is called a subset of B if each element of A is an element of B, and it is expressed as A ⊂ B. The collection of odd integers is a subset of itself.

One approved method of forming a set is to consider a property P possessed by certain elements of a given set X. The set of elements of X having property P may be considered as a set Y. The expression p ε X is used to denote the fact that p is an element of X. Then Y = {p | p ε X and p has property P}. Another approved method is to consider the set Z of all subsets of a given set X. Paradoxically, it is not permissible to regard the collection of all sets as a set.

In set theory, one is interested not only in the properties of sets but also in operations involving sets: addition, subtraction, multiplication, and mapping. The sum of A and B (A + B or A ∪ B) is the set of all elements in either A or B; that is, A + B = {p p ε A or p ε B). The intersection of A and B (A · B, A ∩ B, or AB) is the set of all elements in both A and B; that is, A · B = {p p ε A and p ε B). If there is no element which is in both A and B, one says that A does not intersect B and writes A · B = 0. The expression A − B is used to denote the collection of elements of A that do not belong to B; that is A − B = {p p ε A and p ε B}.

The branch of mathematics or logic that is concerned with sets of objects and rules for their manipulation. UNION, INTERSECT and COMPLEMENT are its three primary operations and they are used in relational databases as follows.

Given a file of Americans and a file of golfers, UNION would create a file of all Americans and golfers. INTERSECT would create a file of American golfers, and COMPLEMENT would create a file of golfers who are not Americans, or of Americans who are not golfers. See fuzzy logic.

Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its terminology and concepts. These include the operations of union (È), and intersection (Ç). The union of two sets is a set containing all the elements of both sets, each listed once. The intersection is the set of all elements common to both original sets. Set theory is useful in analyzing difficult concepts in mathematics and logic. It was placed on a firm theoretical footing by Georg Cantor, who discovered the value of clearly formulated sets in the analysis of problems in symbolic logic and number theory.

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The modern theory of sets was largely inspired by Cantor, whose proof that the set of real numbers could not be put into a one-to-one correspondence with the set of natural numbers opened the door to the set-theoretic hierarchy, and to the study of transfinite numbers. The first proper axiomatization of the theory was that of Ernst Zermelo (1871-1953) in 1908. The axiomatization followed intense controversy over the nature of the set-theoretic hierarchy, the legitimacy of the axiom of choice, and the right approach to the paradoxes lying at the centre of naïve views about sets, of which the best known is Russell's paradox. Classical set theory uses the axiomatization of Zermelo, augmented by the axiom of replacement due to A. Fraenkel (1891-1965). It has been shown that this is equivalent to a natural ‘iterative’ conception, whereby starting with the empty set or null set, and forming only sets of sets, the entire set-theoretic hierarchy can be generated. Philosophically set theory is central because sets are the purest mathematical objects, and it is known that the rest of mathematics can be formulated within set theory (so that numbers, relations, and functions all become particular sets). Particular topics within set theory are indexed under their own headings.

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