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By Maxwell Rosenlicht. Set theory is the language of mathematics. The most complicated ideas in modern mathematics are developed in terms of the basic notions of set theory.
Fortunately the grammar and vocabulary of set theory are extremely simple, at least in the sense that it is possible to go very far in mathematics with only a small amount of set theory.
It so happens that the subject of set theory not only underlies mathematics but has become itself an extensive branch of study; however we do not enter deeply into this study because there is no need to. All we must do here is familiarize ourselves with some of the basic ideas so that the language may be used with precision. A first reading of this chapter can be very rapid since it is mainly a matter of getting used to a few words.
There is occasional verbosity, directed toward the clarification of certain simple ideas which are really somewhat more subtle than they appear. We do not attempt to define the word set. Intuitively a set is a collection, or aggregate, or family, or ensemble all of which words are used synonymously with set of objects which are called the elements, or members of the set, and the set is completely determined by the knowledge of which objects are elements of it. We may speak, for example, of the set of students at a certain university; the elements of this set are the individual students there.
Similarly we may speak of the set of all real numbers to be discussed in some detail in the next chapter , or the set of all straight lines in a given plane, etc. It should be noted that the elements of a set may themselves be sets; for example each element of the set of all straight lines in a given plane is a set of points, and we may also consider such less mathematical examples as the set of married couples in a given town, or the set of regiments in an army.
We shall generally use capital letters to denote sets and lower-case letters to denote their elements. The statement " x is not an element of S " is abbreviated. In the same way there is a difference between a university class consisting of one student and the student himself, or between a committee consisting of one person and that person. The above notation however is not always feasible. A more frequently used notation is. Thus if R is the set of real numbers,. If X and Y are sets and every element of X is also an element of Y , we say that X is a subset of Y ; this is written.
The empty set is the set with no elements. A source of confusion to beginners is that although the empty set contains nothing, it itself is something namely some particular set, the one characterized by the fact that nothing is in it. In a similar way, when dealing with numbers, say with ordinary integers, we must be careful not to regard the number zero as nothing: zero is something, a particular number, which represents the number of things in nothing.
Note that for any set X we have. In symbols,. That is. The word or is used here in the manner that is standard in mathematics. In ordinary language the word or is often exclusive, that is, if A and B are statements, then " A or B is understood to mean A or B but not both", whereas in mathematics it always means " A or B, or both A and B".
If X is a subset of a set S, then the complement of X in S is the set of all elements of S which are not elements of X. If it is explicitly stated, or clear from the context, exactly what the set S is, we often omit the words "in S" for the complement of X. These operations are illustrated in Figure 1, where the sets in question are sets of points in plane regions bounded by curves. For example, if X and Y are subsets of a set S, then. It must be shown that the two sets have the same elements, in other words that each element of the set on the left is an element of the set on the right and vice versa.
This completes the proof. If X and Y are sets, the notation X — Y. Thus if X and Y are subsets of some set S,. Two sets are said to be disjoint if they have no element in common. A collection of any number of sets is said to be disjoint if every two of the sets are disjoint. The intersection and union of more than two sets may be defined in an obvious manner. For example, if X , Y , Z are sets then. More generally the intersection and union of arbitrary families of sets may be defined, and in an obvious way.
The only problem is finding an adequate notation for an arbitrary family of sets, and this is done as follows. The set of all sets Xi as i ranges over I is denoted. It must be shown that each element of the set on the left is an element of the set on the right, and vice versa.
If a and b are objects, by the ordered pair a, b we mean the two objects a and b in a definite order, a first, b second. Ordinary rectangular coordinates in the plane give the usual pictorial representation of the cartesian product: the whole plane can be identified with the product of the two coordinate axes. In Figure 3 there is a more complicated picture in which X, Y are subsets of the two coordinate axes and the cartesian product is a subset of the first quadrant.
If X and Y are sets, by a function from X to Y or a function from X into Y , or a function on X with values in Y is meant a rule which associates with each element of X a definite element of Y. The word mapping, or map, is often used instead of function. The rule can be given in many ways, some of which are discussed below, but the essential thing is that given any element of X there is associated, somehow, some definite element of Y. Two functions from X to Y are considered equal if and only if both functions associate with each specific element of X the same element of Y.
Functions are usually denoted by small letters, such as f. The statement " f is a function from X to Y" is often written. We say that f sends x into f x , or that f maps x into f x , or that x and f x correspond under f. One way, which is usually not very practical, is to list all the elements of X, listing with each one the corresponding one of Y. Or the rule may be given by a mathematical formula.
For example, if X and Y are both taken to be the set R of real numbers, an equation like. Finally we remark that the rule defining a function need not be practically computable.
For example, for x any real number, let f x denote that integer 0,1, …, 9 which is in the billionth decimal place of x to be precise, since a real number x may have more than one decimal representation, as in 1. It is useful to note that the word function alone can be defined in primitive terms, not only the more complete concept "function from X into Y ": a function is an ordered pair whose first member is an ordered pair of sets, say X , Y , and whose second member is a function from X into Y.
For many purposes it is important to bear this fact in mind, but most often we do not make any explicit mental note of it. The composed function is usually denoted g o f, so that we have. Note that what goes under the name "the function x " in elementary calculus is actually the identity function on the set of real numbers.
The two uses we have made for the symbol f are related by the equation. This set, together with the various ideas associated with it, such as its ordering the fact that its elements can be written down in a definite order , or such as the fact that two of its elements may be added to obtain a third with certain general rules holding for this addition, can be obtained from the primitive principles of set theory.
In this section we shall for convenience assume a few simple facts about the natural numbers in order to get as quickly as possible to certain other easy matters of set theory. However all the facts about the set of natural numbers that are used here will be proved explicitly in the next chapter. The notions developed in this section will not be applied until later, so no circular reasoning occurs.
Thus a set is finite if we can count its elements and run out of elements after we count a certain number, say n, of them. The number n depends only on the set X , not on the order in which its elements are counted off; n is called the number of elements in X. For Completeness we say that the number of elements in the empty set is zero. Any subset of a finite set is itself finite, and if it is a proper subset it has a smaller number of elements. Upload Sign In Join. Home Books Science.
Create a List. Download to App. Length: pages 4 hours. Description This well-written text provides excellent instruction in basic real analysis, giving a solid foundation for direct entry into advanced work in such fields as complex analysis, differential equations, integration theory, and general topology. The nominal prerequisite is a year of calculus, but actually nothing is assumed other than the axioms of the real number system. Because of its clarity, simplicity of exposition, and stress on easier examples, this material is accessible to a wide range of students, of both mathematics and other fields.
Chapter headings include notions from set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, interchange of limit operations, the method of successive approximations, partial differentiation, and multiple integrals. Following some introductory material on very basic set theory and the deduction of the most important properties of the real number system from its axioms, Professor Rosenlicht gets to the heart of the book: a rigorous and carefully presented discussion of metric spaces and continuous functions, including such topics as open and closed sets, limits and continuity, and convergent sequence of points and of functions.
Subsequent chapters cover smoothly and efficiently the relevant aspects of elementary calculus together with several somewhat more advanced subjects, such as multivariable calculus and existence theorems.
The exercises include both easy problems and more difficult ones, interesting examples and counter examples, and a number of more advanced results.
Introduction to Analysis lends itself to a one- or two-quarter or one-semester course at the undergraduate level. It grew out of a course given at Berkeley since Splines and Variational Methods Author P.
Invitation to Geometry Author Z. Dennis Lawrence. Analytic Inequalities Author Nicholas D. Abelian Varieties Author Serge Lang. Related Categories. Sets are sometimes indicated by listing all their members between braces.
That is These operations are illustrated in Figure 1, where the sets in question are sets of points in plane regions bounded by curves. Intersection, union, and complement. A proof of this formula is given below. For example, if X , Y , Z are sets then and Clearly , and similarly for the union of three sets.
The set of all sets Xi as i ranges over I is denoted and the intersection and union of this family of sets, together with their respective conventional symbols, are defined by EXERCISE. Cartesian product.
Introduction to Analysis
The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. This can be thought of either as a brief introduction to real analysis, or as a rigorous calculus book: it proves nearly all the facts that are used in single- and multi-variable calculus, but generally does not go beyond that to the more general problems considered in real analysis. This is a Dover unaltered reprint of the edition from Scott, Foresman. The book does cover some topics that are usually not touched on in calculus, such as some theory of complete metric spaces, uniform convergence and uniform continuity, and the inverse and implicit function theorems. Unlike most calculus courses, everything is done analytically rather than geometrically; for example, the transcendental functions are defined by power series.
By Maxwell Rosenlicht. Set theory is the language of mathematics. The most complicated ideas in modern mathematics are developed in terms of the basic notions of set theory. Fortunately the grammar and vocabulary of set theory are extremely simple, at least in the sense that it is possible to go very far in mathematics with only a small amount of set theory.
Standard topics of an introductory analysis course will be covered, along with additional concepts which do not necessarily follow the text. Attendence is required and the exams will be over the lectures and homework. The course topics include: Countable and uncountable sets, the real numbers, order, least upper bounds, and the Archimedean property. Metric spaces: topology, open and closed sets, convergent sequences, completeness, compactness and the Heine-Borel Theorem for the real line, connectedness.