Comninos Mathematical and Computer Programming Techniques for Computer Graphics

1. Auflage 2010
ISBN: 978-1-84628-292-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 548 Seiten

Provides a comprehensive and detailed coverage of the fundamentals of programming techniques for computer graphics Uses lots of code examples, encouraging the reader to explore and experiment with data and computer programs (in the C programming language)

Autoren/Hrsg.

Comninos, Peter

Leseproben

"1 Set Theory Survival Kit (p. 3-4)

Unlike many other branches of mathematics, where the formulation of ideas and concepts occurs gradually over time and is developed by many mathematicians before it is formalised into a single theory, the formulation of set theory is almost the single-handed creation of one mathematician, namely Georg Cantor. Georg Ferdinand Ludwig Philipp Cantor (1845–1918) was born in Russia to a Danish father and a Russian mother and spent most of his life in Germany.

Between the years 1879 and 1884 Cantor published a six-part treatise on set theory (where he introduced some of the fundamental notions of this theory) followed by the publication of a two-part treatise between the years 1895 and 1897 (where he clarified and systematised what he had introduced in his first cycle of publications). Between the years 1897 and 1902 a number of paradoxes in Cantor’s set theory began to emerge. These paradoxes were discovered by Cantor himself and, among others, by the Italian mathematician Cesare Burali-Forti (1861–1931), the German mathematician Ernst Friedrich Ferdinand Zermelo (1871–1953) and the British mathematician Bertrand Arthur William Russell (1872–1970).

In 1908, Zermelo was the first to attempt to introduce an axiomatic approach to the study of set theory. Since then, many mathematicians proved influential in the further development of set theory. Among these are the German mathematician Adolf Abraham Halevi Fraenkel (1891–1965), the Hungarian mathematician and computer scientist John von Neumann (1903–1957), the Swiss mathematician Paul Isaac Bernays (1888–1977) and the Czech mathematician Kurt G¨odel (1906– 1978).

Since its introduction, set theory has proved to be of great importance to the modern formulation of many topics of pure mathematics. In current mathematical practice, such topics as numbers, relations, intervals, functions and transformations are defined in terms of sets. In our study of computer graphics we will frequently use sets to explain a number of other mathematical concepts. Thus, it is important to gain a good understanding of sets and set theory.

1.1 Some Basic Notations and Definitions

1.1.1 Sets and Elements

The concept of the set is one of the basic concepts of mathematics and is fundamental to most branches of modern mathematics. Thus, we start our discussion by defining the terms set and element or member. A set is any well-defined list, collection or class of objects, in which the order and multiplicity of these objects has no significance and is ignored. These objects are called the elements or members of the set. The phrase well-defined means that there is a clear and unambiguous way of defining the elements of a set, i.e. of determining if a given element is a member of a given set. Sets may be finite or infinite depending on the number of their elements. Set theory is the branch of mathematics that concerns the study of sets and their properties."