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Appendix Review of Prerequisites Appendix Review of Prerequisites The most important prerequisite for studying this book is a thorough grounding in advanced calculus. Since there are hundreds of books that treat this subject well, we will simply assume familiarity with it, and re- mind the reader of important facts when necessary. We also assume that the reader is familiar with the terminology and rules of ordinary logic. The other prerequisites are a solid understanding of the basic properties of sets, metric spaces, and groups, at the level that you would find in most undergraduate courses in real analysis and abstract algebra. In this appendix we briefly review some fundamental aspects of these three subjects. If you have not studied this material before, you cannot hope to learn it from scratch here. But this appendix can serve as a reminder of important concepts that you may have forgotten, as a way to standardize our notation and terminology, and as a source of references to books where you can look up more of the details to refresh your memory. You can use the exercises to test your knowledge, or to brush up on any aspects of the subject on which you feel your knowledge is shaky. Set Theory In this book, as in most modern mathematics, mathematical statements are couched in the language of set theory. We give here a brief descriptive summary of the parts of set theory that we will use, in the form that is commonly called “naive set theory.” The word naive should be understood 338 Appendix: Review of Prerequisites in the same sense in which it is used by Paul Halmos in his classic text Naive Set Theory [Hal74]: The axioms of set theory are to be viewed much as Euclid viewed his geometric axioms, as intuitively clear statements of fact from which reliable conclusions can be drawn. One must be a bit careful with the axioms, to be sure, because it is possible to get into trouble by trying to construct sets too freely, as is illustrated by the famous paradox of Bertrand Russell described below. It is primarily for this reason that we take the trouble to enumerate the axioms at all. For more detail on the subject, in the same spirit as the treatment here, consult [Hal74] or [Dev93]. We leave it to the set theorists to explore the deep consequences of the axioms and the relationships among different axiom systems. Basic Concepts The word set is, mathematically, an undefined term. A set should be thought of as an assemblage of “mathematical objects,” whatever they may be—things such as numbers, ordered pairs, functions, or other sets. The properties of sets, and the rules for manipulating them, are expressed in the axioms we list below. We sometimes use the words collection and family as synonyms for set. The fundamental relationship involving sets, which we also leave math- ematically undefined, is that of membership. Intuitively, if x is one of the objects in the set S, then we say that x is a member or an element of S, or x belongs to S, written x ∈ S. The essential characteristic of sets is that they are determined by their members. Formally, we define S = T to mean x ∈ S ⇐⇒ x ∈ T . The set containing no elements is called the empty set and denoted by ∅. It is unique, because any two sets with no elements are equal by our definition of set equality, so we are justified in calling it “the” empty set. (We could postulate its existence as a separate axiom, but its existence will follow from our other axioms, as you will see below.) If S and T are sets such that every element of S is also an element of T , then S is a subset of T , written S ⊂ T .Itisaproper subset if S ⊂ T but S = T . The notation T ⊃ S means S ⊂ T . Clearly, S = T if and only if S ⊂ T and T ⊂ S. The axioms for sets describe precisely what sets can be asserted to exist, and what properties they have. Here is the first one. • Specification axiom: Given a set S and a sentence P (x) that is either true or false whenever x is any particular element of S, there is a set consisting of all those x ∈ S for which P (x) is true, denoted by {x ∈ S : P (x)}. Note that one must start with a specific set before the specification axiom can be used. This requirement rules out forming sets out of self- contradictory specifications such as the one discovered by Bertrand Russell Set Theory 339 and now known as “Russell’s paradox”: The sentence C = {X : X ∈ X} looks as if it might define a set, but it does not, because each statement C ∈ C and C ∈ C implies its own negation. Similarly, the specification axiom implies that there does not exist a “set of all sets,” for if there were such a set S, we could use the specification axiom to define C = {S ∈ S : S ∈ S} and reach the same contradiction. Still, there are times when we will need to speak of “all sets” or other similar aggregations, primarily in the context of category theory (see Chap- ter 7). For this purpose, we reserve the word class to refer to an aggregate of mathematical objects that may or may not constitute a set. • Power set axiom: Given any set S, there is a set P(S), called the power set of S, whose elements are exactly the subsets of S. • Union axiom: Given any collection C of sets, there is a set called their union and denoted by C, with the property that x ∈ C if and only if x ∈ S for some S ∈ C. Given any nonempty collection C of sets, their intersection, denoted by C, is defined as the set C = {x ∈ C : x ∈ S for every S ∈ C}. Other notations for unions and intersections are S; S1 ∪ S2 ∪··· ; ∈C S S; S1 ∩ S2 ∩··· . S∈C Given any collection C of sets, if A∩B = ∅ whenever A, B ∈ C and A = B, the sets in C are said to be disjoint. If A and B are any sets, their set difference is defined to be the set A B = {x ∈ A : x ∈ B}, which exists by the specification axiom. If B ⊂ A, the set difference A B is also called the complement of B in A. When sets are defined by specification, it is common to abbreviate the notation in certain circumstances if it can be done unambiguously. For ex- ample, if the elements of a set can be named explicitly, the set is commonly specified simply by listing its elements, as in {a1,a2,...,ak}. As long as each of the elements ai is an element of some other set Si, this is a legit- imate use of our axioms and can be interpreted as {x ∈ S1 ∪···∪Sk : x = a1 or x = a2 or ... or x = ak}. Since the resulting set is the same regardless of what sets Si the ai’s originally came from, there is no need to include them in the notation. A set {a} with a unique element a is called a singleton. 340 Appendix: Review of Prerequisites Cartesian Products, Relations, and Functions Another primitive concept that we will use without a formal definition is that of an ordered pair. Think of it as a pair of objects in a specific order, indicated by writing them in parentheses and separated by a comma, as in (a, b). The objects a and b are called the components of the ordered pair. The defining characteristic is that two ordered pairs are equal if and only if their first components are equal and their second components are equal: (a, b)=(a,b) ⇐⇒ a = a and b = b. • Cartesian product axiom: Given sets A and B, there exists a set A × B, called their Cartesian product, whose members are precisely the ordered pairs (a, b) for every a ∈ A and b ∈ B. With these axioms we can define the most important constructions in mathematics: relations and functions. A relation between sets X and Y is a subset of X × Y .Ifr is a relation, it is often convenient to use some notation such as x r y to mean (x, y) ∈ r. For example, both “equals” and “less than” are relations in R × R. An important special case arises when we consider relations between a set S and itself, which we usually call a relation “on S.” Let ∼ denote such a relation. It is said to be reflexive if x ∼ x for all x ∈ S, symmetric if x ∼ y implies y ∼ x, and transitive if x ∼ y and y ∼ z imply x ∼ z.A relation that is reflexive, symmetric, and transitive is called an equivalence relation. Given an equivalence relation ∼, for each x ∈ S the equivalence class of x is defined to be the set [x]={y ∈ S : y ∼ x}. The set of equivalence classes is denoted by S/∼. Closely related to equivalence relations is the following notion: A parti- tion of a set S is a collection C of disjoint nonempty subsets of S whose union is S. In this situation one also says that S is the disjoint union of the sets in C. Exercise A.1. Given an equivalence relation ∼ on a set S, show that the set S/∼ of equivalence classes is a partition of S. Conversely, given a partition of S, show that there is a unique equivalence relation whose set of equivalence classes is exactly the original partition.
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