The intersection of two sets is the set containing the elements common to the two sets and is denoted by the symbol [symbol].

The union of two sets is the set containing all elements belonging to either one of the sets or to both, denoted by the symbol [symbol].

The intersection of a set and its complement is the empty set (denoted by [symbol]), or A[symbol]A´=[symbol]; the union of a set and its complement is the universal set, or A[symbol]A´=U.

www.bartleby.com /65/se/set.html (548 words)

set - HighBeam Encyclopedia(Site not responding. Last check: 2007-10-20)

Membership in a set is indicated by the symbol ∈ and nonmembership by ∉; thus, x ∈ A means that element x is a member of the set A (read simply as "x is a member of A") and y ∉ A means y is not a member of A.

The intersection of two sets is the set containing the elements common to the two sets and is denoted by the symbol ∩.

The union of two sets is the set containing all elements belonging to either one of the sets or to both, denoted by the symbol ∪.

Who's To Blame?(Site not responding. Last check: 2007-10-20)

Setmathematics was developed and refined in the second half of the nineteenth century by mathematicians and philosophers such as Georg Cantor, Guiseppe Peano and Bertrand Russell.

They came to realize that poorly understood mathematical ideas would be clarified if they could be expressed in terms of certain simpler, universal, unambiguous terms and symbols, and they also realized that the concepts from setmathematics provided these basic building blocks from which all of mathematics could be rigorously established.

By defining an infinite set to be any set that has the same cardinality as one of its proper subsets, he was able to develop an impressive theory of infinite numbers, which included the discovery that in fact there are many different levels of infinity.

In his paper, "Mathematics as a Science of Patterns: Ontology and Reference", he states his purpose as developing a philosophy of mathematics in which the logical forms of mathematical statements are taken on face value, i.e., the numerical expressions are singular terms that refer.

This is due to the fact that set theory is extensional, and the combinatorial aspects of mathematics, which is concerned with the finitely presented properties of the inscriptions of the formal language is intensional.

Set theory strips away structure from the ontology of mathematics leaving pluralities of structureless individuals open to the imposition of new structure.

The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics.

Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.

For instance, it is desirable to have the “set of all integers that are divisible by number 3,” the “set of all straight lines in the Euclidean plane that are parallel to a given line”, the “set of all continuous real functions of two real variables” etc.

Category Theoretic Perspectives on the Foundations of Mathematics(Site not responding. Last check: 2007-10-20)

One step further is to use a set theory with a "Universe" (Mac Lane [MacLane71] also discusses use of a universe, and in Cohn's Universal Algebra a set theory is adopted in which every set is a member of a universe, giving a hierarchy of universes).

When you found mathematics in set theory, what you do is take a universe in which there are lots of sets, and use these as the raw material in which to do mathematics.

A debate on "set theory versus category theory" (though that characterisation of the debate is also disputed) has raged on the FOM mailing list, and can be observed in the archives.

Gödel was born in 1906 in the Austro-Hungarian province of Moravia.

Set theory, algebra, analysis – indeed the whole of mathematics is incomplete, assuming that the axioms are finitely given and consistent (which are conditions mathematics cannot forego), and robust enough to account for arithmetic.

This collection (the constructible set universe), is a model of set theory, that is, it supports all of the axioms of set theory (the axioms are “true” in it), and in addition it supports the GCH.

Set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as primary school in many countries.

Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived.

Sets are conventionally denoted with capital letters, A, B, C, etc. Two sets A and B are said to be equal, written A = B, if they have the same members.

The Math Forum - Math Library - Set Theory(Site not responding. Last check: 2007-10-20)

A tutorial on sets, convering the definition of sets and their elements, union, intersection, subsets, and sets of numbers.

Challenging word problems requiring "a sound knowledge of high school mathematics (up to, but not including, calculus) and a fair amount of mathematical insight and ingenuity." Questions in part one are multiple choice; questions in part two are open-ended.

Mathematical Logic - Department of Mathematics, Manchester University, U.K. There is a strong tradition in mathematical logic at Manchester.

This report presents a discussion of the mathematics character repertoire of the Unicode Standard [Unicode] as used for mathematics, but it is intended that this discussion apply to mathematical notation in general.

Mathematical Markup Language (MathML™) [MathML], an XML application [XML], is a major beneficiary of the increased repertoire for mathematical symbols.

The current set of fonts for use in the character code charts was prepared after consultation with the American Mathematical Society and leading publishers of mathematics, and shows much simpler forms that are derived from the forms written on a flboard.

In mathematics, an uncountable or nondenumerable set is a set which is not countable.

The diagonalization proof technique can also be used to show that several other sets are uncountable as well, for instance the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of natural numbers.

If there exists a one-to-one mapping from a set A into set B, then the cardinal number of A is no greater than the cardinal number of B. If there also exists a one-to-one mapping from B into A, then the cardinality of A and B are equal.

Then these two sets are in a sense EQUIVALENT if mapped elements in B behave 'the same way' as unmapped elements in A. Such a correspondence between the sets that respects the pattern of behaviour of combined elements in each set reflects similar or equivalent intrinsic structure between sets.

In the case of "Coyoteman IsoMorph", set A consists of the set of points on the man's face and set B is the set of all points on the surface of the Coyote's face.

The Hub is an Internetworked resource for mathematics and science education funded by the Eisenhower Regional Consortia and operated by TERC on behalf of the Regional Alliance for Mathematics and Science Education Reform.

Mathematical Mayhem is a non-profit mathematical journal written by and for high school and undergraduate university students.

Mathematics in Context (MiC) is an NSF-funded, comprehensive middle-school mathematics curriculum for grades 5-8 that reflects the philosophy and pedagogy of the NCTM standards.

Department of Mathematics: Mandelbrot Set(Site not responding. Last check: 2007-10-20)

An excellent introduction to the mathematics of the Mandelbrot set and its relationship with the associated Julia sets.

These are mainly images of parts of the Mandelbrot set, designed to entice you to buy the Fractal eXtreme fractal exploration program.

It allows you to choose what part of M you wish to zoom in on, and how much you want to zoom in; as a bonus, the associated Julia set for the central point in the image can also be graphed.