16 I I : Basic concepts This unit is the beginning of the strictly mathematical development of set theory in the course. We begin with a brief discussion of how mathematics is written and continue with a summary of the main points in logic that arise in mathematics. The latter is mainly meant as background and review, and also as a reference for a few symbols that are frequently used as abbreviations. In the remaining sections we introduce the most essential notions of set theory and some of their simplest logical interrelationships. Mathematical language Mathematicians are like Frenchmen; whatever you say to them they translate into their own language and forthwith it is something entirely different. J. W. von Goethe (1749 – 1832) A page of mathematical writing is different from a page of everyday writing in many respects, and for an inexperienced or uninitiated reader it is often more difficult to understand. Before considering strictly mathematical topics in these notes, it might be helpful to summarize some special features of mathematical language and the reasons for such differences. The language of mathematics is a special case of technical language or language for special purposes. As such, it has many things in common with other specialized language uses in the other sciences and also in legal writing. In all these contexts, it is important to state things precisely and to justify assertions based upon earlier writing. It is also important to avoid things which are unrelated to the substance of the discussion, including emotional appeals and nearly all personal remarks; when the latter appear, they are usually restricted to a small part of the text. The need for precise, impersonal language affects mathematical writing in several ways. We shall list some notable features below. 1. Sentences tend to be long and carefully written, sometimes at the expense of clarity. This is often necessary to avoid misunderstandings. For example, in mathematics when one divides a number x by a number y, it is necessary to stipulate that y be nonzero. 2. In scientific writing there is more of a tendency to stress nouns and modifiers rather than verbs, and there is a much greater use of the passive voice. For example, instead of saying, “You can do X,” one generally sees the more17 impersonal, “It is possible to do X.” This reinforces the unimportance or anonymity of the individual who does X. However, a reader who is not used to such an impersonal style might view it as uninviting. 3. Precise meanings must be attached to specific words. These do not necessarily correspond to a word’s everyday meaning(s), and of course there are also many words that are rarely if ever seen elsewhere. Words like “product” and “set” and “differentiate” are examples of words whose mathematical meanings differ from standard usage. Other words such as “abelian” or “eigenvector” or “integrand” are essentially unique to mathematics and only appear when mathematics is presented or applied to another subject. 4. There is an extensive use of references to the writings of others. Such citations are logically indispensable and make everything more concise, but they can also make it difficult or impossible to read through something without frequent interruptions. 5. Particularly in the sciences, there is a heavy reliance on symbols such as numerals, operators (for example, the plus and equals signs), formulas or equations, and diagrams as well as other graphics. These allow the writer to express many things quickly but precisely. However, they may be difficult to decipher, particularly for a beginner. The pros and cons of mathematical (and other scientific) language are reflected by a surprising fact: Even though such material is more difficult to read than an ordinary book, it is much easier to translate scientific writings to or from a foreign language than it is to translate a best seller or a regular column in a newspaper. In particular, adequate computerized translations of scientific articles are considerably easier to produce than acceptable computerized translations of literature. Both clarity and preciseness are important in mathematical (and other scientific) writing. A lack of precision can lead to costly mistakes in scientific experiments and engineering projects (similar considerations apply to legal writing, where ambiguities involving simple words can lead to extensive and expensive litigation). On the other hand, a lack of clarity can undermine the fundamental goals of communicating information. Every subject has tried to adopt guidelines for balancing these contrasting aims, but probably there will always be challenges to doing so effectively in all cases. I I .0 : Topics from logic (Lipschutz, §§ 10.1 – 10.12) Mathematics is based upon logical principles, and therefore some understanding of logic is required to read and write mathematics correctly. In this course we shall take the most basic concepts of logic for granted. Our main purpose here is to describe the key logical points and symbolic logical notation that will be used more or less explicitly in this course. Chapter 10 of Lipschutz contains numerous examples illustrating the main points of logic that we shall use in this course, and it it provides additional background18 and reference material. Sections 1.1 – 1.5 of Rosen also treat these topics in an introductory but systematic manner. In most mathematical writings, the logical arguments are carried out using ordinary language and standard algebraic symbolism. When logical terminology as developed in this section is used, it is often used intermittently for purposes of abbreviation when ordinary wording becomes too lengthy or awkward; there are similarities between this and the practice of explaining some programming issues in a pseudo – code that is halfway between ordinary and computer language. Although such logical abbreviations are only used sometimes in mathematics, it is important to be familiar with them and recognize them when they do appear. Concepts from propositional calculus The basic objects in propositional calculus are simple declarative sentences, and by convention each sentence is either true or false. There are several simple grammatical and logical operations that can be used to connect sentences. 1. If P and Q are sentences, then the sentence P and Q is