We give a discussion of some basic mathematical concepts and terminologies connected with sets, numbers, functions and graphs.

This chapter considers the ways in which Puccini’s works were staged during his lifetime, with a close focus upon the composer’s own vision of how his operas should look. Puccini was keenly involved with the visual representation of his works on stage and always strove to be as ‘authentic’ as possible in reproducing the locales being depicted. This chapter draws upon the composer’s correspondence to demonstrate how proactive Puccini was in every aspect of the performance of his works. We find Puccini complaining about stage designs for his works and suggesting improvements, as well as suggesting novel scenographic approaches to make a work more effective for the viewer. Puccini was also extremely keen to maintain aspects of staging and direction used in some of the source plays upon which his operas had been based. The chapter concludes with a discussion of the various extant sources that document Puccini’s staging intentions.

In this appendix, we collect together some basic material that readers should know before studying this book. The material includes sets, enumeration, and probability. In practice, we have found that students have often already been exposed to this material and only need a quick review, so we did not want to include another introductory chapter delaying the study of our main topics. We cover sets, subsets, universal sets, unions,intersections and complements, Cartesian products, power sets, and rules such as De Morgan's laws. For enumeration, we cover the addition and multiplications rules and special cases such as permutations and combinations. For probability, although our coverage is less detailed than other chapters, we actually cover more than we need for the rest of the book, including conditional probability and Bayes's theorem. Most readers will be comfortable just using this chapter as a reference or review.

In this chapter, we deconstruct the computable general equilibrium model and describe its core elements. These include sets, endogenous and exogenous variables, exogenous parameters, behavioral and identity equations, and model closure. We describe composite commodities and prices, price relationshipds, price normalization, price transmission, and the numeraire. We explain how the CGE model runs and how to carry out an experiment.

The aim of this chapter is twofold. First, we recall a collection of basic mathematical notions that are needed for the discussions of the following chapters. Second, we have a first, still purely mathematical, look at the central topics of the book: languages, relations and functions between strings, as well as important operations on languages, relations and functions. We also introduce monoids, a class of algebraic structures that gives an abstract view on strings, languages, and relations.