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An Introduction to Random Sets by Hung T. Nguyen (Chapman & Hall/CRC) Random sets as models for set-valued observations, are a new type of data proving useful in areas such as survey sampling, biostatistics, and intelligent systems. This is the first text to explore the topic in depth, using extended probability theory to provide a framework and tools for statistical analysis of random sets. With an abundance of examples, it highlights the basic role random sets play in a variety of statistical settings, links their study to fuzzy logic, fully develops the theory, and concludes with a variety of applications. Written by an author of the best-selling A First Course in Fuzzy Logic, this book is rigorous yet readable and fills a significant need for a textbook treatment of the subject.

The study of random sets is a large and rapidly growing area with connections to many areas of mathematics and applications in widely varying disciplines, from economics and decision theory to biostatistics and image analysis. The drawback to such diversity is that the research reports are scattered throughout the literature, with the result that in science and engineering, and even in the statistics community, the topic is not well known and much of the enormous potential of random sets remains untapped.

An Introduction to Random Sets provides a friendly but solid initiation into the theory of random sets. It builds the foundation for studying random set data, which, viewed as imprecise or incomplete observations, are ubiquitous in today's technological society. The author, widely known for his best-selling A First Course in Fuzzy Logic text as well as his pioneering work in random sets, explores motivations, such as coarse data analysis and uncertainty analysis in intelligent systems, for studying random sets as stochastic models. Other topics include random closed sets, related uncertainty measures, the Choquet integral, the convergence of capacity functionals, and the statistical framework for set-valued observations. An abundance of examples and exercises reinforces the concepts discussed.

Designed as a textbook for a course at the advanced undergraduate or beginning graduate level, this book will serve equally well for self-study and as a reference for researchers in fields such as statistics, mathematics, engineering, and computer science.

This text is designed for a one-semester course at the advanced undergraduate or beginning graduate level. It is also intended for use as a reference book for researchers in fields such as probability and statistics, artificial intelligence, computer science, engineering. It is a friendly but solid introduction to the topic of random sets for those who need a strong background for further study. After completing the course, the students should be able to read more specialized and advanced books on the subject as well as articles in technical and professional journals.

The material presented in this text is drawn from many sources in the literature, including our own research. The presentation of the material is from the ground up. The prerequisite consists simply of a good upper-level undergraduate course in probability and statistics. A summary of concepts and results in probability theory is given in the Appendix.

The theory of random sets is viewed as a natural generalization of probability and statistics on random vectors, i.e., of multivariate statistical analysis. Random set data can be also viewed as imprecise/incomplete observations which are frequent in today's technological societies. As models for set-valued observations as well as for the process underlying the gathering of perception-based information, via coarsening schemes, random sets are a new type of data. As such, new mathematical tools for statistical inference and decision making need to be developed. In the foreword to Mathéron's book on random sets [73], G. Watson expressed his vision of statistics as follows: "Modern statistics must be defined as the applications of computers and mathematics to data analysis. It must grow as new types of data are considered and as computing technology advances."

The Structure of Proof: With Logic and Set Theory by Michael L. O'Leary (Prentice Hall) There are many approaches that one can take with regard to a course dedi­cated to teaching proof writing. Some prefer to teach the mathematics and the structure of the proofs simultaneously. Others choose to teach the methods of proof and then apply those methods to various topics. This is the strategy of this text. Here we boldly jump into a discussion of logic and examine some details with the belief that if the details are not understood well, then the ap­plication of those details will suffer later. Moreover, it is the understanding of how logic interacts with mathematics that empowers the student to have the courage and confidence to tackle greater problems in courses such as Abstract Algebra or Topology. How we accomplish this is outlined below.

TEXT OUTLINE

This text is designed for a one-semester course on the fundamentals of proof writing for students with a modest calculus background. The text is divided into three parts: Logical Foundations, Main Topics, and Coming Attractions.

Part I: Logical Foundations. Since it is a requirement for any proof, the text begins with an introduction to mathematical logic. This part begins by studying sentences with and without variables and concludes by writing basic paragraph‑style proofs.

1. Propositional Logic. In this chapter we translate propositions using logical symbols and translate the symbols into English. Connectives and truth tables are covered, and formal two‑column proofs are in­troduced. These proofs require students to carefully follow the rules of logic and serve as a model for paragraph proofs.

2. Predicates and Proofs. Here we cover basic sets, quantification, and negations of quantifiers. Two-column proofs are written using propositional forms with quantifiers. Strategies that are covered in­clude Direct and Indirect Proof, Biconditional Proof, and Proof by Cases. A transition to writing paragraph-style proofs is included throughout.

Part II: Main Topics. The logic covered in the first part can be viewed as an advanced organizer for writing proofs. This is best seen in the first application of Part II: set theory. This is a natural first choice, for set theory is just one step from logic. Sets are found throughout the part as we study induction, well‑ordered sets, congruence classes, relations, equiva­lence classes, and functions.

3. Set Theory. The basic set operations as well as inclusion and equal­ity are covered. Two sections are devoted to families of sets and operations with them.

4. Mathematical Induction. Various forms of mathematical induction are studied. Applications include combinatorics, recursion, and the Well‑Ordering Principle. Under recursion we look at the Fibonacci sequence and (just for fun!) the golden ratio makes an appearance.

5. Number Theory. We take a look at the axioms of number theory and discuss topics such as divisibility, greatest common divisors, primes, and congruences.

6. Relations and Functions. As a prelude to our look at functions, re­lations are examined. The main example is the equivalence relation. Our look at functions includes the notions of well‑defined, domain, range, one‑to‑one, onto, image, pre‑image, and cardinality.

Part III: Coming Attractions. The topics seen in this last part will be­come familiar to the student of mathematics. Even so, logic and sets are not forgotten. We see these subjects at work when we study rings and then move from the discrete to the continuous and study topology.

7. Ring Theory. The study of ring theory is viewed as a generalization of the study of the integers. We encounter integral domains, fields, subrings, ideals, factors, homomorphisms, and polynomials.

8. Topology. The text closes with a look at the various spaces and subsets that are important to topology. Topics include metric spaces, normed spaces, open and closed sets, isometries, and limits.

As this overview suggests, the text contains more topics than can be cov­ered during a semester. This provides greater flexibility for the instructor, and it gives the student a one‑stop reference on the basics of undergraduate math­ematics and the proof structures needed for success. This will benefit mathe­matics students as they take their upper level courses in algebra and analysis. It will also benefit mathematics education students who must teach many of the fundamental concepts.

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