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Unformatted text preview: MATHEMATICAL BACKGROUND This supplementary chapter (present on the CDROM version of the book only) describes some of the mathematical concepts behind the graphical techniques introduced in chapter 32 of ObjectOriented Software Construction, second edition . It is extracted from the ISE manual on EiffelBuild [M 1995e] . The conventions, and any crossreference that you may encounter in this chapter, are those of [M 1995e] rather than the rest of ObjectOriented Software Construction . Pages are numbered 1076.1, 1076.2 and so on to avoid any confusion with the page numbers of the rest of the book, as they appear in the printed version. 1076.2 This page intentionally blank 1076.3 This page intentionally blank 32A Mathematical background 32A.1 OVERVIEW EiffelBuild relies on simple properties of functions. This chapter presents a summary of the necessary notions. You can use EiffelBuild without having read this discussion , and in fact if you are eager to get your hands on EiffelBuild you may prefer to skip this chapter on first reading and move immediately to the following chapter and the Guided Tour. But an understanding of the elementary mathematical notions discussed below will help you get the most out of EiffelBuild , especially for advanced uses. Many of the topics of this chapter are also useful for the formal study of programming languages, and are covered in more details in the book Introduction to the Theory of Programming Languages . 32A.2 FINITE SETS , CARTESIAN PRODUCT A finite set may be given by the list of its members in braces , for example PERSON { Hlne , Kiyoko , Laura , Roberto , Helmut } COUNTRY { Japan , France , Italy , UK } where means is defined as. A note about naming conventions : the example sets used in this chapter , such as PERSON and COUNTRY , follow the Eiffel rules for types ( classes ): they are written in uppercase letters , and use the singular rather than the plural. So PERSON denotes a set of persons and COUNTRY a set of countries. A mathematical text might call these sets PEOPLE and COUNTRIES , but for a programmer it is more attractive to think of declarations of the form Hlne : PERSON ( Eiffel syntax ) meaning Hlne represents an object of type PERSON , hence the singular. If X and Y are sets , then X Y , called the cartesian product of these sets , is the set of all pairs of the form < x , y > where x is a member of X and y is a member of Y . For example the set PERSON COUNTRY contains all the pairs such as < Hlne , Japan >, < Hlne , France >, ... , < Kiyoko , Japan >, < Kiyoko , France > and so on. Cartesian product is also applicable to infinite sets. For example if N is the set of natural ( nonnegative ) integers, then N N is the set of all possible pairs of natural integers....
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This note was uploaded on 10/02/2009 for the course CS 4376 taught by Professor Christeansan during the Spring '09 term at Dallas Colleges.
 Spring '09
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