A Practical Introduction to Data Structures and Algorithm Analysis

2Mathematical PreliminariesThis chapter presents mathematical notation, background, and techniques usedthroughout the book. This material is provided primarily for review and reference.You might wish to return to the relevant sections when you encounter unfamiliarnotation or mathematical techniques in later chapters.Section 2.7 on estimating might be unfamiliar to many readers. Estimating isnot a mathematical technique, but rather a general engineering skill. It is enormouslyuseful to computer scientists doing design work, because any proposedsolution whose estimated resource requirements fall well outside the problem’s resourceconstraints can be discarded immediately.2.1 Sets and RelationsThe concept of a set in the mathematical sense has wide application in computerscience. The notations and techniques of set theory are commonly used when describingand implementing algorithms because the abstractions associated with setsoften help to clarify and simplify algorithm design.A set is a collection of distinguishable members or elements. The membersare typically drawn from some larger population known as the base type. Eachmember of a set is either a primitive element of the base type or is a set itself.There is no concept of duplication in a set. Each value from the base type is eitherin the set or not in the set. For example, a set named P might be the three integers 7,11, and 42. In this case, P’s members are 7, 11, and 42, and the base type is integer.Figure 2.1 shows the symbols commonly used to express sets and their relationships.Here are some examples of this notation in use. First define two sets, Pand Q.P = {2, 3, 5}, Q = {5, 10}.25