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Unformatted text preview: 2 C H A P T E R Basic Structures: Sets, Functions, Sequences, and Sums 2.1 Sets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations M uch of discrete mathematics is devoted to the study of discrete structures, used to represent discrete objects. Many important discrete structures are built using sets, which are collections of objects. Among the discrete structures built from sets are combinations, unordered collections of objects used extensively in counting; relations, sets of ordered pairs that represent relationships between objects; graphs, sets of vertices and edges that connect vertices; and finite state machines, used to model computing machines. These are some of the topics we will study in later chapters. The concept of a function is extremely important in discrete mathematics. A function assigns to each element of a set exactly one element of a set. Functions play important roles throughout discrete mathematics. They are used to represent the computational complexity of algorithms, to study the size of sets, to count objects, and in a myriad of other ways. Useful structures such as sequences and strings are special types of functions. In this chapter, we will introduce the notion of a sequence, which represents ordered lists of elements. We will introduce some important types of sequences, and we will address the problem of identifying a pattern for the terms of a sequence from its first few terms. Using the notion of a sequence, we will define what it means for a set to be countable, namely, that we can list all the elements of the set in a sequence. In our study of discrete mathematics, we will often add consecutive terms of a sequence of numbers. Because adding terms from a sequence, as well as other indexed sets of numbers, is such a common occurrence, a special notation has been developed for adding such terms. In this section, we will introduce the notation used to express summations. We will develop formulae for certain types of summations. Such summations appear throughout the study of discrete mathematics, as, for instance, when we analyze the number of steps a procedure uses to sort a list of numbers into increasing order. 2.1 Sets Introduction In this section, we study the fundamental discrete structure on which all other discrete structures are built, namely, the set. Sets are used to group objects together. Often, the objects in a set have similar properties. For instance, all the students who are currently enrolled in your school make up a set. Likewise, all the students currently taking a course in discrete mathematics at any school make up a set. In addition, those students enrolled in your school who are taking a course in discrete mathematics form a set that can be obtained by taking the elements common to the first two collections. The language of sets is a means to study such collections in an organized fashion. We now provide a definition of a set. This definition is an intuitive definition, which is not part of a formal theory of sets.not part of a formal theory of sets....
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- Spring '08