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Appendix-A-Sets-Functions

Appendix-A-Sets-Functions - Appendix A Sets and Functions...

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Appendix A Sets and Functions This appendix establishes the notation of sets and functions as used in the book. We review the use of this mathematical language to describe sets in a variety of ways, to combine sets using the operations of union, intersection, and product, and to derive logical consequences. We also review how to formulate and understand predicates, and we define certain sets that occur frequently in the study of signals and systems. Finally, we review functions. A.1 Sets A set is a collection of elements . The set of natural numbers , for example, is the collection of all positive integers. This set is denoted (identified) by the name Naturals , Naturals = { 1 , 2 , 3 , ···} . (A.1) In ( A.1 ) the left hand side is the name of the set and the right hand side is an enumeration or list of all the elements of the set. We read ( A.1 ) as ‘ Naturals is the set consisting of the numbers 1 , 2 , 3 , and so on.’ The ellipsis ‘ ··· ’ means ‘and so on’. Because Naturals is an infinite set we cannot enumerate all its elements, and so we have to use ellipsis. For a finite set, too, we may use ellipsis as a convenient shorthand as in A = { 1 , 2 , 3 , ··· , 100 } , (A.2) which defines A to be the set consisting of the first 100 natural numbers. Naturals and A are sets of numbers. The concept of sets is very general, as the examples below illustrate. Students is the set of all students in this class, each element of which is referenced by a student’s name: Students = { John Brown , Jane Doe , Jie Xin Zhou , ···} . USCities consists of all cities in the U.S., referenced by name: USCities = { Albuquerque , San Francisco , New York , ···} . 513
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514 APPENDIX A. SETS AND FUNCTIONS BooksInLib comprises all books in the U.C. Berkeley library, referenced by a 4-tuple (first author’s last name, book title, publisher, year of publication): BooksInLib = { (Lee, Digital Communication , Kluwer, 1994) , (Walrand, Communication Networks , MorganKaufmann, 1996) , ···} . BookFiles consists of all L A T E X documents for this book, referenced by their file name: BookFiles = { sets.tex , functions.tex , ···} . We usually use either italicized, capitalized names for sets, such as Reals and Integers , or single capital letters, such as A , B , X , Y . An element of a set is also said to be a member of the set. The number 10 is a member of the set A defined in ( A.2 ), but the number 110 is not a member of A . We express these two facts by the two expressions: 10 A , 110 / A . The symbol ‘ ’ is read ‘is a member of’ or ‘belongs to’ and the symbol ‘ / ’ is read ‘is not a member of’ or ‘does not belong to.’ When we define a set by enumerating or listing all its elements, we enclose the list in curly braces {···} . It is not always necessary to give the set a name. We can instead refer to it directly by enumerating its elements. Thus { 1 , 2 , 3 , 4 , 5 } is the set consisting of the numbers 1 , 2 , ··· , 5.
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