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    Set Theory and Logic
    Set Theory and Logic
    Set Theory and Logic
    Ebook754 pages12 hours

    Set Theory and Logic

    By Robert R. Stoll

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    About this ebook

    Set Theory and Logic is the result of a course of lectures for advanced undergraduates, developed at Oberlin College for the purpose of introducing students to the conceptual foundations of mathematics. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. One of the most complex and essential of modern mathematical innovations, the theory of sets (crucial to quantum mechanics and other sciences), is introduced in a most careful concept manner, aiming for the maximum in clarity and stimulation for further study in set logic.
    Contents include: Sets and Relations — Cantor's concept of a set, etc.
    Natural Number Sequence — Zorn's Lemma, etc.
    Extension of Natural Numbers to Real Numbers
    Logic — the Statement and Predicate Calculus, etc.
    Informal Axiomatic Mathematics
    Boolean Algebra Informal Axiomatic Set Theory Several Algebraic Theories — Rings, Integral Domains, Fields, etc.
    First-Order Theories — Metamathematics, etc.
    Symbolic logic does not figure significantly until the final chapter. The main theme of the book is mathematics as a system seen through the elaboration of real numbers; set theory and logic are seen s efficient tools in constructing axioms necessary to the system.
    Mathematics students at the undergraduate level, and those who seek a rigorous but not unnecessarily technical introduction to mathematical concepts, will welcome the return to print of this most lucid work.
    "Professor Stoll . . . has given us one of the best introductory texts we have seen." — Cosmos.
    "In the reviewer's opinion, this is an excellent book, and in addition to its use as a textbook (it contains a wealth of exercises and examples) can be recommended to all who wish an introduction to mathematical logic less technical than standard treatises (to which it can also serve as preliminary reading)." — Mathematical Reviews.
    LanguageEnglish
    PublisherDover Publications
    Release dateMay 23, 2012
    ISBN9780486139647
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    • 4/5
      A concise treatment of the subject, but with enough explanation to be suitable for self-study. My main complaint: no solutions or hints for the exercises. The older (?) style notation also tripped me up in places.

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    Set Theory and Logic - Robert R. Stoll

    LOGIC

    CHAPTER 1

    Sets and Relations

    T HE THEORY OF SETS as a mathematical discipline originated with the German mathematician G. Cantor (1845-1918). A complete account of its birth and childhood is out of the question here, since a considerable knowledge of mathematics is a prerequisite for its comprehension. Instead, we adopt the uneasy compromise of a brief sketch of these matters. If this proves too difficult for the reader, nothing is lost; on the other hand, if it is at least partially understood, something may be gained.

    Cantor’s investigation of questions pertaining to trigonometric series and series of real numbers led him to recognize the need for a means of comparing the magnitude of infinite sets of numbers. To cope with this problem, he introduced the notion of the power (or size) of a set by defining two sets as having the same power if the members of one can be paired with those of the other. Since two finite sets can be paired if and only if they have the same number of members, the power of a finite set may be identified with a counting number. Thus the notion of power for infinite sets provides a generalization of everyday counting numbers. Cantor developed the theory, including an arithmetic, of these generalized (or transfinite) numbers and in so doing created a theory of sets. His accomplishments in this area are regarded as an outstanding example of mathematical creativity.

    Cantor’s insistence on dealing with the infinite as an actuality—he regarded infinite sets and transfinite numbers as being on a par with finite sets and counting numbers—was an innovation at that time. Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers. Other applications in analysis and geometry were found, and Cantor’s theory of sets won acceptance to the extent that by 1890 it was recognized as an autonomous branch of mathematics. About the turn of the century there was some change in attitude with the discovery that contradictions could be derived within the theory. That these were not regarded as serious defects is suggested by their being called paradoxes— defects which could be resolved, once full understanding was acquired. The problems posed by Cantor’s theory, together with its usefulness, gradually created independent interest in a general theory of abstract sets in which his ideas appeared in greatly extended form. That general theory forms the basis of this chapter.

    Specifically, this chapter discusses, within the framework of set theory, three important mathematical concepts: function, equivalence relation, and ordering relation. Sections 3-6 contain the necessary preliminaries, and Sections 1 and 2 describe our point of departure for Cantor’s theory.

    One might question the wisdom of choosing a starting point which is known to lead ultimately to disaster. However, we contend that the important items of this chapter are independent of those features which characterize the Cantorian or naive approach to set theory. Indeed, any theory of sets, if it is to serve as a basis for mathematics, will include the principal definitions and theorems appearing in this chapter. Only the methods we employ to obtain some of these results are naive. No irreparable harm results in using such methods; they are standard tools in mathematics.

    In this chapter we assume that the reader is familiar with the systems of integers, rational numbers, real numbers, and complex numbers. Knowledge in these areas will enlarge the possibilities for constructing examples to assist the assimilation of definitions, theorems, and so on. We shall reserve the underlined letters Z,Q,R, and C for the sets of integers, rational numbers, real numbers, and complex numbers, respectively, and the symbols Z+, Q+, and R+ for the sets of positive integers, positive rationals, and positive reals, respectively.

    1. Cantor’s Concept of a Set

    Let us consider Cantor’s concept of the term set and then analyze briefly its constituent parts. According to his definition, a set S is any collection of definite, distinguishable objects of our intuition or of our intellect to be conceived as a whole. The objects are called the elements or members of S.

    The essential point of Cantor’s concept is that a collection of objects is to be regarded as a single entity (to be conceived as a whole). The transfer of attention from individual objects to collections of individual objects as entities is commonplace, as evidenced by the presence in our language of such words as bunch, covey, pride, and flock.

    With regard to the objects which may be allowed in a set, the phrase objects of our intuition or of our intellect gives considerable freedom. First, it gives complete liberty so far as the nature of the objects comprising a set is concerned. Green apples, grains of sand, or prime numbers are admissible constituents of sets. However, for mathematical applications it is reasonable to choose as members such mathematical entities as points, lines, numbers, sets of numbers, and so on. Second, it permits the consideration of sets whose members cannot, for one reason or another, be explicitly exhibited. In this connection one is likely to think first of infinite sets, for which it is not even theoretically possible to collect the members as an assembled totality. The set of all prime numbers and the set of all points of the Euclidean plane having rational coordinates in a given coordinate system are examples of this. On the other hand, there are finite sets which display the same degree of intangibility as any infinite set.

    An old example which serves to bear out this contention begins with the premise that a typesetting machine with 10,000 characters (these would include the lower-case and capital letters of existing alphabets in various sizes of type, numerals, punctuation, and a blank character for spacing) would be adequate for printing in any language. (The exact size of the set of characters is not at issue; the reader may substitute for 10,000 any integer greater than 1.) Let it be agreed that by a book is meant a printed assemblage of 1,000,000 characters, including blank spaces. Thus a book may contain from 0 to 1,000,000 actual characters. Now consider the set of all books. Since there are 10,000 possibilities available for each of the 1,000,000 positions in a book, the total number of books is equal to 10,000¹,⁰⁰⁰’⁰⁰⁰. This is a large (but finite!) number. In addition to books of gibberish there would appear in the set all textbooks ever written or planned, all newspapers ever printed, all government pamphlets, all train schedules, all logarithm tables ever computed, and so on, and so on. The magnitude eludes comprehension to the same degree as does that of an infinite set.

    The remaining key words in Cantor’s concept of a set are distinguishable and definite. The intended meaning of the former, as he used it, was this: With regard to any pair of objects qualified to appear as elements of a particular set, one must be able to determine whether they are different or the same. The attribute definite is interpreted as meaning that if given a set and an object, it is possible to determine whether the object is, or is not, a member of the set. The implication is that a set is completely determined by its members.

    2. The Basis of Intuitive Set Theory

    and write

    x A

    if the object x is a member of the set A. If x is not a member of A, we shall write

    x ∉ A.

    Further,

    x1, x2, … , xn A

    will be used as an abbreviation for " xA and xA and … and xn A."

    In terms of the membership relation, Cantor’s assumption that a set is determined by its members may be stated in the following form.

    The intuitive principle of extension. Two sets are equal iff (if and only if) they have the same members.

    The equality of two sets X and Y will be denoted by

    X = Y,

    and the inequality of X and Y by

    X ≠ Y.

    Among the basic properties of this relation are

    X = X,

    X = Y implies Y = X,

    x = Y and Y = Z imply x = Z,

    for all sets X, Y, and Z.

    It should be understood that the principle of extension is a nontrivial assumption about the membership relation. In general, a proof of the oquality of two specified sets A and B is in two parts: one part demonstrates that if x A, then x B ; the other demonstrates that if x B, then x A. An example of such a proof is given below.

    That (uniquely determined) set whose members are the objects x1 x2 … , xn will be written

    {x1, x2 … , xn}.

    In particular, {x}, a so-called unit set, is the set whose sole member is x.

    EXAMPLES

    2.1. Let us prove that the set A of all positive even integers is equal to the set B of positive integers which are expressible as the sum of two positive odd integers. First we assume that x A and deduce that x B. If x A, then x = 2m, and hence x = (2m BB AB, then x = (2p – 1) + (2q – 1), and hence x = 2(p + q – 1), which implies that x A. Thus, we have proved that A and B have the same members.

    2.2.  {2, 4, 6} is the set consisting of the first three positive even integers. Since {2, 4, 6} and {2, 6, 4} have the same members, they are equal sets. Moreover, {2, 4, 6} = {2, 4, 4, 6} for the same reason.

    2.3.  The members of a set may themselves be sets. For instance, the geographical area known as the United States of America is a set of 50 member states, each of which, in turn, is a set of counties (except Alaska, which has boroughs). Again, {{1,3}, {2, 4}, {5, 6}} is a set with three members, namely, {1,3}, {2, 4}, and {5, 6}. The sets {{1, 2}, {2, 3}} and {1, 2, 3} are unequal, since the former has {1, 2} and {2, 3} as members, and the latter has 1, 2, and 3 as members.

    2.4.  The sets {{1, 2}} and {1, 2} are unequal, since the former, a unit set, has {1, 2} as its sole member and the latter has 1 and 2 as its members. This is an illustration of the general remark that an object and the set whose sole member is that object are distinct from each other.

    We digress briefly to comment on the alphabets which we shall employ in discussing set theory. Usually, lower-case italic English letters will denote elements, and, for the time being, capital italic letters will denote sets which contain them. Later, lower-case Greek letters will be introduced for a certain type of set. If the members of a set are themselves sets, and if this is noteworthy in the discussion, capital script letters will be used for the containing set, and it will be called a collection of sets. of all finite sets A of integers x. As a rule of thumb, the level of a set within a hierarchy of sets under consideration is suggested by the size and gaudiness of the letter employed to denote it.

    Although the brace notation is practical for explicitly defining sets made up of a few elements, it is too unwieldly for defining sets having a large, finite number of elements and useless for infinite sets (sets having infinitely many elements). How can sets with a large number of elements be described? In this connection one instinctively tends to differentiate between finite and infinite sets on the grounds that a finite set can be realized as an assembled totality whereas an infinite set cannot. However, a large finite set (for example, the set of books described in Section 1) is as incapable of comprehension as is any infinite set. On the basis of such examples one must conclude that the problem of how to describe efficiently a large finite set and the problem of how to describe an infinite set are, for all practical purposes, one and the same.

    A commonly accepted solution, devised by Cantor, is based on the concept of a "formula in x." At this time we offer only the following intuitive description. Let us understand by a statement a declarative sentence capable of being classified as either true or false. Then, by a formula in x we understand a finite sequence made up from words and the symbol x such that when each occurrence of x is replaced by the same name of an object of an appropriate nature a statement results. For instance, each of the following is a formula in x:

    5 divides x;      x² + x + 1 > x;

    x loves John;     x² = 2.

    x < x;

    In contrast, neither of the following is a formula in x:

    for all x, x² – 4 = (x – 2)(x + 2);

    there is an x such that x² ≤ 0.

    Rather, each is simply a statement. A grammarian might describe a formula in x, alternatively, as a sentence which asserts something about x. Clearly, each sentence of the first list above has this quality, whereas neither of the second list has. A still different approach to this concept is by way of the notion of function as it is used in elementary mathematics. A formula in x may be described as a function of one variable such that for a suitable domain the function values are statements.

    We shall use a capital English letter followed by the symbol (x) to denote a formula in x. If, in a given context, P(x) stands for a particular formula, then P(a) stands for the same formula with a in place of x.

    Our objective, that of describing sets in terms of formulas, is achieved by way of the acceptance of the following principle.

    The intuitive principle of abstraction. A formula P(x) defines a set A by the convention that the members of A are exactly those objects a such that P(a) is a true statement.

    Because sets having the same members are equal, a given formula determines exactly one set which, in mathematics, is usually denoted by

    {x|P(x)},

    read "the set of all x such that P(x)." Thus a {x|P(x)} iff P(a) is a true statement. It may be said that the decision as to whether a given object a is a member of {x|P(x)} is that of whether a possesses a certain property (or quality). Because of this, when a formula in x, P(x), is applied to a set construction it is commonly called a property of x and, indeed, the defining property of {x|P(x)}. Further, our principle of abstraction is then described by the assertion that every property determines a set.

    We shall admit the possibility of the occurrence of symbols other than x in a formula in x. If P(x) is a formula in x and y is a symbol that does not occur in P(x), then, as properties, P(x) and P(y) are indistinguishable, and so {x|P(x)} = {y|P(y)}. This need not be the case, however, if y does occur in P(x). For example,

    { x|x is divisible by u } = { y|y is divisible by u },

    but

    {x|x is divisible by y} ≠ {y|y is divisible by y}.

    Again, if F(x) and G(x) are two properties such that F(x) holds for x when and only when G(x) holds for x, then {x|F(x)} = {x|G(x)}, by an application of the principle of extension. For example,

    {x | x A and x B) = {x|x B and x A},

    and

    {x | x Z+ and x < 5} = {xZ+ and (x + l)² ≤ 29}.

    EXAMPLES

    2.5.  The introduction of infinite sets by defining properties is a familiar procedure to a student of analytic geometry. One need merely recall the customary definition of such geometric loci as the conic sections. For instance, the circle of radius 2 centered at the origin is the set of all x such that x is a point in the plane and at a distance of two units from the origin.

    2.6.   The following are examples of easily recognized sets defined by properties.

    (a)  {x|x is an integer greater than 1 and having no divisors less than or equal to x½}.

    (b)  {x|x is a positive integer less than 9}.

    (c)  {x|x is a line of slope 3 in a coordinate plane}.

    (d)  {x|x is a continuous function on the closed interval from 0 to 1}.

    2.7. {x|x = x1 or x = x2 or … or x = xn} is the set we earlier agreed to denote by {x1, x2,…, xn}.

    2.8. In some cases our language makes possible, by way of a property, a briefer definition of a finite set than can be achieved by an enumeration of the elements. For example, it is shorter to define a particular set of 100 people by the property "x is a senator" than by enumerating names of the members.

    2.9. If A is a set, then x A is a formula in x and may be used as a defining property of a set. Since y {x|x A} iff y A, we have

    A = {x|x A},

    by virtue of the principle of extension.

    Various modifications of the basic brace notation for sets are used. For example, it is customary to write

    {x A|P(x)}

    instead of {x|x A and P(x)} for the set of all objects which are both members of A and have property P(x). An alternative description of this set is "all members of A which have property P(x)," and it is this description that the new notation emphasizes. As illustrations, {x R | 0 ≤ x ≤ 1} denotes the set of all real numbers between 0 and 1, inclusive, and {x Q+, |x² < 2} denotes the set of all positive rationals whose square is less than 2.

    If P(x) is a property and f is a function, then

    {f(x)| P(x)}

    will be used to denote the set of all y for which there is an x such that x has property P(x) and y = f(x). For example, instead of writing

    {y| there is an x such that x is an integer and y = 2x }

    we shall write

    {2x|x Z}.

    Again, {x²|x x, y † of real numbers x and y, it is reasonable to interpret {(x, yR² |y = 2x} as the line through the origin having slope 2.

    The principle of set extension, the principle of abstraction, and the principle of choice (which will not be formulated until there is need for it) constitute the working basis of Cantor’s theory of sets. It is of interest to note that although we made an attempt, prior to introducing the first two principles, to describe what a set is, neither of these principles nor the third includes a definition of the word set. Rather, each is merely an assumption about sets. The basic concept used to enunciate these principles is membership. Consequently, the membership relation for sets, rather than the notion of set itself, assumes the role of the principal concept of set theory.

    We have already mentioned that contradictions can be derived within intuitive set theory. The source of trouble is the unrestricted use of the principle of abstraction. Of the known contradictions the simplest to describe is that discovered by Bertrand Russell in 1901. It is associated with the set R having the formula x ∉ x as its defining property and may be stated as: On one hand, R R, and on the other hand, R ∉ R. The reader can easily supply informal proofs of these two contradictory statements.

    EXERCISES

    {1, 2, 3}.

    {{1, 2, 3}, {1, 3}, 1, 2}? Justify your answer.

    2.3.Try to devise a set which is a member of itself.

    2.4.Give an example of sets A, B, and C such that A B, B C, and A ∉ C.

    2.5.Describe in prose each of the following sets.

    (a) {x Z|x is divisible by 2 and x is divisible by 3}.

    (b) {x|x A and x B}.

    (c) {x|x B}.

    Z| for some integer y, x = 2y} and x {x Z| for some integer y, x = 3y}}.

    (e) {x²|x is a prime}.

    (f) {a/b Q |a + b = 1 and a, b Q}.

    x,y) R²|x²+y² = 1}.

    (h) {(x, yR² | y = 2x and y = 3x}.

    2.6. Prove that if a, b, c, and d are any objects, not necessarily distinct from one another, then {{a}, {a, b}} = {{c}, {c, d)} iff both a = c and b = d.

    3.Inclusion

    We now introduce two further relations for sets. If A and B are sets, then A is included in B, symbolized by

    A ⊆ B,

    iff each member of A is a member of B. .In this event one also says that A is a subset of B. Further, we agree that B includes A, symbolized by

    B⊇A,

    is synonymous with A is included in B. Thus, A ⊆ B and B ⊇ A each means that, for all x, if x A, then x B. The set A is properly included in B, symbolized by

    A ⊂ B

    (or, alternatively, A is a proper subset of B, and B properly includes A), iff A⊆ B and A ≠B. For example, the set of even integers is properly included in the set Z of integers, and the set Q of rational numbers properly includes Z.

    Among the basic properties of the inclusion relation are distinct pair of its member

    X ⊆ X;

    X ⊆ Y and Y ⊆ Z imply X ⊆ Z;

    X ⊆ Y and Y ⊆ Z imply X = Z.

    The last of these is the formulation, in terms of the inclusion relation, of the two steps in a proof of the equality of two sets. That is, to prove that x = Y, one proves that X ⊆ Y and then that Y ⊆ X.

    For the relation of proper inclusion, only the analogue of the second property above is valid. The proof that X ⊂ Y and Y ⊂ Z imply X ⊂ Z is required in one of the exercises at the end of this section. There the reader will also find further properties of proper inclusion, so far as its relationship to inclusion is concerned.

    Since beginners tend to confuse the relations of membership and inclusion, we shall take every opportunity to point out distinctions. At this time we note that the analogues for membership of the first two of the above properties for inclusion are false. For example, if X is the set of prime numbers, then X∉ X. {Z}, since Z is the sole member of {Z}.

    We turn now to a discussion of the subsets of a set, that is, the sets included in a set. This is our first example of an important procedure in set theory—the formation of new sets from an existing set. The principle of abstraction may be used to define subsets of a given set. Indeed, if P(x) is a formula in x and A is a set, then the formula

    A and P(x)

    determines that subset of A which we have already agreed to write as{x A |P(x)}. If A is a set and we choose P(x) to be x ≠ x, the result is{ x A|x ≠ x } and this set, clearly, has no elements. The principle of extension implies that there can be only one set with no elements. We call this set the empty set and symbolize it by

    ∅.

    The empty set is a subset of every set. To establish this it must be proved that if A is a set, then each member of ∅ is a member of A. Since ∅ has no members, the condition is automatically fulfilled. Although this reasoning is correct, it may not be satisfying. An alternative proof which might be more comforting is an indirect one. Assume that it is false that ∅ ⊆ A. This can be the case only if there exists some member of ∅ which is not a member of A. But this is impossible, since ∅ has no members. Hence, ∅ ⊆ A is not false; that is, ∅ ⊆ A.

    Each set A ≠ ∅ has at least two distinct subsets, A and ∅. Moreover, each member of A determines a subset of A; if a A, then {a} ⊆ A. There are occasions when one wishes to speak not of individual subsets of a set, but of the set of all subsets of that set. The set of all subsets of a set A is the power set of A, symbolized by

    (A).

    (A) is an abbreviation for

    {B|B ⊆ A}.

    For instance, if A = {1, 2, 3}, then

    (A) = {A, {1,2}, {1,3}, {2,3}, {1}, {2}, {3}, ∅}.

    As another instance of the distinction between the membership and inclusion relations we note that if B ⊆ A, then B (A), and if a A, then {a} ⊆ A and {a(A).

    The name "power set of A" for the set of all subsets of A has its origin in the case where A (A) has 2n members if A has n members. To prove this, consider the following scheme for describing a subset B of A = {a1 … , an} : a sequence of n 0’s and l’s where the first entry is 1 if aB and 0 if a1 ∉ B and where the second entry is 1 if a2 B and 0 if a2∉ B , and so on. Clearly, the subsets of A can be paired with the set of all such sequences of 0’s and l’s; for example, if n = 4, then {a1, a3(A) is equal to 2n.

    EXERCISES

    3.1. Prove each of the following, using any properties of numbers that may be needed.

    (a) {x Z} for an integer y, x = 6y} = {x Z| for integers u and v, x = 2u and x = 3v}.

    (b) {x R| for a real number y, x = y²} = {x R|x ≥ 0}.

    Z for an integer y, x = 6y} ⊆ {x Z | for an integer y, x = 2y}.

    3.2. Prove each of the following for sets A,B, and C.

    (a) if A ⊆ B and B ⊆ C, then A ⊆ C

    (b) If A ⊆ B and B ⊂ C, then A ⊂ C.

    (c) If A ⊂ B and B ⊆ C, then A ⊂ C.

    (d) If A ⊂B and B ⊂ C, then A ⊂ C.

    3.3.  Give an example of sets A, B,C,D, and E which satisfy the following conditions simultaneously: A ⊂ B, B C, then C ⊂ D, and D ⊂ E.

    3.4. Which of the following are true for all sets A, B, and C?

    (a) If A ∉ B and B ∉ C, then A ∉ C.

    (b) If A ≠ B and B ≠ C, then A ≠ C.

    (c) If A B and B C, then A∉ C.

    (d). If A⊂ B and B ⊂ C then C A

    (e) If A ⊆ B and B C, then A ∉ C.

    3.5.  Show that for every set A, A ⊆ ∅ iff A = ∅.

    3.6.  Let A1, A2, …,An be n sets show that

    A1 ⊆ A2 ⊆ …, ⊆ An ⊆ A1 iff A1 = A2 = … = An

    3.7. Give several examples of a set X such that each element of X is a subset of X.

    (A) if A = {{1, 2}, {3}, 1}.

    3.9. For each positive integer n, give an example of a set An of n elements such that for each pair of elements of An, one member is an element of the other.

    4.Operations for Sets

    We continue with our description of methods for generating new sets from existing sets by defining two methods for composing pairs of sets. These so-called operations for sets parallel, in certain respects, the familiar operations of addition and multiplication for integers. The union (sum, join) of the sets A and B, symbolized by A ∪ B and read " A union B" or" A cup B," is the set of all objects which are members of either A or B; that is,

    A ∪ B = {x|x A or x B}.

    Here the inclusive sense of the word or is intended. Thus, by defnition, x A ∪ B iff x is a member of at least one of A and B. For example,

    {1,2,3} ∪ (1,3,4} = {1,2, 3, 4}.

    The intersection (product, meet) of the sets A and B, symbolized by A ∩ B and read "A intersection B" or "A cap B," is the set of all objects which are members of both A and B; that is,

    A ∩ B = {x | x A and x B}

    Thus, by definition, x A ∩ B iff x A and x B. For example,

    {1,2,3} ∩ {1,3,4} = {1,3}.

    It is left as an exercise to prove that for every pair of sets A and B the following inclusions hold:

    ∅ ⊆ A ∩ B ⊆ A ⊆ A ∪ B.

    Two sets A and B are disjoint iff A ∩ B = ∅, and they intersect iff A ∩ B ≠ ∅ . A collection of sets is a disjoint collection iff each shidistinct pair of its member sets is disjoint. A partition of a set X of nonempty and distinct subsets of X such that each member of X For example, {{1,2}, {3}, {4,5}} is a partition of {1,2,3,4,5}.

    A further procedure, that of complementation, for generating sets from existing sets employs a single set. The absolute complement of a set A, symbolized by

    is {x|x∉ A). The relative complement of A with respect to a set X is X ; this is usually shortened to x – A, read "X minus A." Thus

    x X|x ∉ A },

    that is, the set of those members of X which are not members of A. The symmetric difference of sets A and B, symbolized by A + B, is defined as follows:

    A + B = (A - B) ∪ (B - A).

    This operation is commutative, that is, A + B = B + A, and associative, that is, (A + B) + C = A + {B + C). Further, A + A = ∅, and A + ∅ = A. Proofs of these statements are left as exercises.

    If all sets under consideration in a certain discussion are subsets of a set U, then U is called the universal set (for that discussion). As examples, in elementary number theory the universal set is Z, and in plane analytic geometry the universal set is the set of all ordered pairs of real numbers. A graphic device known as a Venn diagram is used for assisting one’s thinking on complex relations which may exist among subsets of a universal set U. A Venn diagram is a schematic representation of sets by sets of points: the universal set U is represented by the points within a rectangle, and a subset A of U is represented by the interior of a circle or some other simple region within the rectangle. The complement of A relative to Uwithout confusion, is the part of the rectangle outside the region representing A, as shown in Figure 1. If the subsets A and B of U are represented in this way, then A ∪ B and A ∩ B are represented by shaded regions, as in Figure 2 and Figure 3, respectively. Disjoint sets are represented by nonoverlapping regions, and inclusion is depicted by displaying one region lying entirely within another. These are the ingredients for constructing the Venn diagram of an expression compounded from several sets by means of union, intersection, complementation, and inclusion. The principal applications of Venn diagrams are to problems of simplifying a given complex expression and simplifying given sets of conditions among several subsets of a universe of discourse. Three simple examples of this sort appear below. In many cases such diagrams are inadequate, but they may be helpful in connection with the algebraic approach developed in the next section.

    Figure 1

    Figure 2

    Figure 3

    EXAMPLES

    4.1. Suppose A and B are given sets such that A – B = B – A =∅. Can the relation of A to B be expressed more simply? Since A —B =∅ means A ∩ =∅, the regions representing A do not overlap (= B, so we conclude (Figure 5) that A ⊆ B. Conversely, if A ⊆ B, it is clear that A – B = ∅. We conclude that A – B = ∅ iff A ⊆ B. Interchanging A and B gives B – A = ∅ iff B ⊆A Thus the given relations hold between A and B iff A ⊆ B and B ⊆ A or, A = B.

    Figure 4

    Figure 5

    Figure 6

    4.2. Let us investigate the question of whether it is possible to find three subsets A, B, and C of ∪ such that

    C≠ ∅, A ∩ B ≠ ∅, A ∩ C= ∅, (A ∩B) - C =.∅

    The second condition implies that A and B intersect and, therefore, incidentally that neither is empty. From Example 4.1 the fourth condistion amounts to A ∩ B ⊆ C, from which it follows that the first is superfluous. The associated Venn diagram indicates that A and C intersect; that is, the validity of the second and fourth conditions contradicts the third. Hence, there do not exist sets satisfying all the conditions simultaneously.

    4.3.Given that F, G, and L are subsets of U such that

    F ⊆ G, G ∩ L ⊆ F, L ∩ F = ∅.

    Is it possible to simplify this set of conditions? The Venn diagram (Figure 6) represents only the first and third conditions. The second condition forces L and G to be disjoint, that is, G ∩ L = ∅. On the other hand, if F ⊆ G and G ∩ L =∅ , then all given conditions hold. Thus F ⊆ G and G ∩ L = ∅ constitute a simplification of the given conditions.

    EXERCISES

    (Note: Venn diagrams are not to be used in Exercises 4.1-4.8.)

    4.1. Prove that for all sets A and B, ∅⊆ A ∩ B ⊆ A ∪ B.

    4.2. Let Z be the universal set, and let

    A = {x Z | for some positive integer y, x = 2y},

    B = {x Z | for some positive integer y, x= 2y — 1},

    C Z | x < 10}.

    and C – (A ∪ B), either in prose or by a defining property.

    4.3. Consider the following subsets of Z+, the set of positive integers:

    A = {x Z+| for some integer y, x = 2y},

    B = {x Z+| for some integer y, x = 2y + 1},

    C = {x Z+|x < 10}.

    (a)   Describe A ∩ C, B ∪ C, and B - C.

    (b).  Verify that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

    4.4. If A is any set, what are each of the following sets? A ∩ ∅, A ∪ ∅,A – A ∅, ∅ – A.

    4.5.Determine ∅ ∩{∅}, {∅} ∩ {∅}, {∅, {∅}} – ∅, {∅, {∅}} – {∅},{∅,{∅}} – {{∅}}

    4.6. Suppose A and B are subsets of U. Show that in each of (a), (b), and (c) below, if any one of the relations stated holds, then each of the others holds.

    (a) A A ∪ B = B, A ∩ B = A.

    (b) A ∩B = ∅, A , B

    (c) A ∪ B ⊆ B⊆ A.

    4.7.  Prove that for all sets A, B, and C,

    (A ∩ B) ∪ C = A ∩ (B ∪) iff C ⊆ A.

    4.8.Prove that for all sets A, B, and C,

    (A – B) – C = (A – C) – (B – C).

    4.9.(a) Draw the Venn diagram of the symmetric difference, A + B, of sets A and B.

    (b) Using a Venn diagram, show that symmetric difference is a commutative and associative operation.

    (c) Show that for every set A, A + A = ∅ and A + ∅ = A.

    4.10. The Venn diagram for subsets A, B, and C of U, in general, divides the rectangle representing U into eight nonoverlapping regions. Label each region with a combination of A, B, and C which represents exactly that region.

    4.11. With the aid of a Venn diagram investigate the validity of each of the following inferences:

    (a) If A, B, and C are subsets of U such that A ∩ Band A ∪ C ⊆ B, then A ∩ C = ∅.

    (b) If A, B, and C are subsets of U such that A and B , then B = ∅

    5. The Algebra of Sets

    If we were to undertake the treatment of problems more complex than those examined above, we would feel the need for more systematic procedures for carrying out calculations with sets related by inclusion, union, intersection, and complementation. That is, what would be called for could appropriately be named the algebra of sets—a development of the basic properties of ∪, ∩, —, and ⊆ together with interrelations. As such, the algebra of sets is intended to be the set- theoretic analogue of the familiar algebra of the real numbers, which is concerned with properties of +, •, and ≤ and their interrelations The basic ingredients of the algebra of sets are various identities—equations which are true whatever the universal set U and no matter what particular subsets the letters (other than U and ∅ ) represent.

    Our first result lists basic properties of union and intersection. For the sake of uniformity, all of these have been formulated for subsets of a universal set U. However, for some of the properties this is a purely artificial restriction, as an examination of the proofs will show.

    THEOREM 5.1.    For any subsets A, B, C of a set U is an abbreviation for U – A.

    Proof.   Each assertion can be verified by showing that the set on either side of the equality sign is included in the set on the other side. As an illustration we shall prove identity 3.

    (a) Proof that A ∪ (B ∩ C)⊆ (A ∪ B) ∩ (A ∪C) Let x A ∪ (B ∩ C). Then x A or x B ∩ C. If x A, then x A ∪ B and x A∪ C, and hence x is a member of their intersection. If x B ∩ C, then x B and x C. Hence x . A ∪ B and x A ∪ C, so again x is a member of their intersection.

    (b) Proof that (A ∪ B) ∩ (A ∪ C) ⊆ A ∪ (B ∩ C). Let x (A ∪B) ∩.(A ∪ C).Then x A ∪ B and x A ∪ C.. Hence x A, or x B and x C. These imply that x A ∪ (B ∩ C).

    Identities 1 and 1′ are referred to as the associative laws for union and intersection, respectively, and identities 2 and 2′ as the commutative laws for these operations. Identities 3 and 3′ are the distributive laws for union and intersection, respectively. The analogy of properties of union and intersection with properties of addition and multiplication, respectively, for numbers, is striking at this point. For instance, 3′ corresponds precisely to the distributive law in arithmetic. That there are also striking differences is illustrated by 3, which has no analogue in arithmetic.

    According to the associative law, identity 1, the two sets that can be formed with the operation of union from sets A, B, and C, in that order, are equal. We agree to denote this set by A ∪ B ∪ C. Then the associative law asserts that it is immaterial as to how parentheses are introduced into this expression. Using induction, this result can be generalized to the following. The sets obtainable from given sets A1, A2, …, An, in that order, by use of the operation of union are all equal to one another. The set defined by A1 A 2, …, An in this way will be written as

    Al ∪ A2 ∪ … ∪ An.

    In view of identity 1′ there is also a corresponding generalization for intersection. With these general associative laws on the record we can state the general commutative law: If 1′, 2′, …, n′ are 1,2, …, n in any order, then

    A1 ∪ A2 ∪ … ∪ An = A1, ∪ A2, ∪ … ∪ An,.

    We can also state the general distributive laws:

    A ∪ (B1 ∩ B2 ∩ … ∩ Bn )

    =( A ∪ B1) ∩ (A ∪ B2) ∩ … ∩ (A ∪ Bn),

    A ∩ (B1 ∪ B2 ∪ … ∪ Bn )

    =(A ∩ B1) ∪ (A ∩ B2) ∪ … ∪ (A ∩ Bn)

    These can also be proved by induction.

    Detailed proofs of the foregoing properties of unions and intersections of sets need make no reference to the membership relation; that is, these properties follow solely from those listed in Theorem 5.1. The same is true of those further properties which appear in the next theorem. Such facts may be regarded as the origin of the axiomatic approach to the algebra of sets developed in Chapter 6. One derivative of this approach is the conclusion that every theorem of the algebra of sets is derivable from 1-5 and l′-5′.

    These ten properties have another interesting consequence. In Theorem 5.1 they are paired in such a way that each member of a pair is obtainable from the other member by interchanging ∪ and ∩ and, simultaneously, ∅ and U. An equation, or an expression, or a statement within the framework of the algebra of sets obtained from another by interchanging ∪ and ∩ along with ∅ and U throughout is the dual of the original. We contend that the dual of any theorem expressible in terms of ∪, ∩, and   –, and which can be proved using only identities 1-5 and 1′-5′, is also a theorem. Indeed, suppose that the proof of such a theorem is written as a sequence of steps and that opposite each step is placed the justification for it. By assumption, each justification is one of 1-5, one of 1′-5′, or a premise of the theorem. Now replace the identity or relation in each step by its dual. Since 1-5 and l′-5′ contain with each its dual, and the dual of each premise of the original theorem is now a premise, the dual of each justification in the original proof is available to serve as a justification for a step in the new sequence which, therefore, constitutes a proof. The last line of the new sequence is, therefore, a theorem, the dual of the original theorem. Accepting the fact that every theorem of the algebra of sets is deducible from 1-5 and l′-5′, we then obtain the principle of duality for the algebra of sets: If T is any theorem expressed in terms of ∪, ∩, and — , then the dual of T is also a theorem. This implies, for instance, that if the unprimed formulas in the next theorem are deduced solely from Theorem 5.1, then the primed formulas follow by duality. The reader should convince himself that all the assertions in Theorem 5.2 are true by using the definitions of ∪, ∩, and — in terms of the membership relation. Further, he might try to deduce some of them solely from Theorem 5.1—that is, without appealing in any way to the membership relation. Some demonstrations of this nature appear in the proof of Theorem 6.2.1. †

    THEOREM 5.2.  For all subsets A and B of a set U the following statements are valid. Here Ā is an abbreviation for U – A.

    6. If, for all A, A ∪ B = A ,    6′. If, for all A, A ∩ B = U, then

       then B = ∅.                           B = U.

    7,7′   If A ∪ B = U and A ∩ B = ∅, then B = Ā.

    = A.

    Some of the identities in Theorem 5.2 have well-established names. For example, 10 and 10′ are the idempotent laws, 12 and 12′ are the absorption laws, and 13 and 13′ the DeMorgan laws. The identities 7, 7′ and 8, 8′ are each numbered twice to emphasize that each is unchanged by the operation which converts it into its dual; such formulas are called self-dual. Note that 7, 7′ asserts that each set has a unique complement.

    A remark about the form of the next theorem is in order. An assertion of the form, "The statements R1, R2, …, Rk are equivalent to one another, means For all i and j , Ri iff Rj, " which, in turn, is the case iff R1 implies R2 implies R3, …, Rk-1 implies Rk and Rk implies R1. The content of the theorem is that the inclusion relation for sets is definable in terms of union as well as in terms of intersection.

    THEOREM 5.3.  The following statements about sets A and B are equivalent to one another.

       (I)  A ⊆ B.

      (II)  A ∩ B = A.

     (III) A ∪ B = B.

    Proof.   (I) implies (II). Assume that A ⊆ B. Since, for all A and B, A ∩ B ⊆ A, it is sufficient to prove that A ⊆ A ∩ B. But if x A, then x B and, hence, x A ∩ B. Hence A ⊆ A ∩ B.

    (II) implies (III). Assume A ∩B = A. Then

    A ∪ B = (A ∩ B) ∪B = (A ∪ B) ∩ ( B ∪ B)

    = (A ∪ B) ∩ B = B.

    (III) implies (I). Assume that A ∪ B = B. Then this and the identity A ⊆ A ∪B imply A ⊆ B.

    The principle of duality as formulated earlier does not apply directly to expressions in which – or ⊆ appears. One can cope with subtraction by using the unabbreviated form, namely, A for A – B. Similarly, by virtue of Theorem 5.3, A ⊆ B may be replaced by A ∩ B = A (or A ∪ B = B). Still better, since the dual of A ∩ B = A is A ∪ B = A, which is equivalent to A ⊇ B, the principle of duality may be extended to include the case where the inclusion symbol is present, by adding the provision that all inclusion signs be reversed.

    EXAMPLES

    5.1.With the aid of the identities now available a great variety of complex expressions involving sets can be simplified, much as in elementary algebra. We give three illustrations.

    5.2. There is a theory of equations for the algebra of sets, and it differs considerably from that encountered in high school algebra. As an illustration we shall discuss a method for solving a single equation in one unknown. Such an equation may be described as one formed using ∩, ∪, and — on symbols A1, A2, … An, and X, where the A’s denote fixed subsets of some universal set U and X denotes a subset of U which is constrained only by the equation in which it appears. Using the algebra of sets, the problem is to determine under what conditions such an equation has a solution and then, assuming these are satisfied, to obtain all solutions. A recipe for this follows; the proof required in each step is left as an exercise (see Exercise 5.7).

    Step I.   Two sets are equal iff their symmetric difference is equal to ∅. Hence, an equation in X is equivalent to one whose righthand side is ∅.

    Step II.   An equation in X with righthand side ∅ is equivalent to one of the form

    (A ∩ X) ∪ (B ) = ∅,

    where A and B are free of X.

    Step III.   The union of two sets is equal to ∅ iff each set is equal to ∅. Hence, the equation in Step II is equivalent to the pair of simultaneous equations

    A ∩ X = ∅, B = ∅.

    Step IV.   The above pair of equations, and hence the original equation, has a solution iff B ⊆ Ā In this event, any X, such that B ⊆ X ⊆ Ā, is a solution.

    We illustrate the foregoing by deriving necessary and sufficient conditions that the following equation have a solution:

    (The introduction of X in the preceding equation is discussed in Exercise 5.7.)

    Thus, the original equation has a solution iff

    It is left as an exercise to show that this condition simplifies to C ⊆D

    EXERCISES

    5.1.Prove that parts 3′, 4′, and 5′ of Theorem 5.1 are identities.

    5.2.Prove the unprimed parts of Theorem 5.2 using the membership relation. Try to prove the same results using only Theorem 5.1. In at least one such proof write out the dual of each step to demonstrate that a proof of the dual results.

    5.3.Using only the identities in Theorems 5.1and 5.2, show that each of the following equations is an identity.

     5.4. Rework Exercise 4.9(b), using solely the algebra of sets developed in this section.

     5.5. Let A1, A2, …, An be sets, and define Sk to be A1 ∪ A2 ∪ … ∪ Ak for k = 1, 2, … n. Show that

    = {A1, A2 – S1, A3 – S2, …, An – S n –1}

    is a disjoint collection of sets and that

    Sn = A1 ∪ (A2 – S1) ∪ … ∪ (An – Sn–l).

    a partition of Sn?

    5.6. Prove that for arbitrary sets A1, A2, …, An(n ≥ 2),

    A1 ∪ A2 ∪ …∪An = (A1 – A2) ∪(A2 – A2) ∪ … ∪(An-1 – An)

    ∪(An –A1) ∪ (A1 ∩A2 ∩ …∩ An )

    5.7. Referring to Example 5.2, prove the following.

    (a)  For all sets A and B, A = B iff A + B =∅.

    (b) An equation in X with righthand member ∅ can be reduced to one of the form ( A ∩ X) ∪ (B ∩ ) = ∅. (Suggestion: Sketch a proof along these lines. First, apply the DeMorgan laws until only complements of individual sets appear. Then expand the resulting lefthand side by the distributive law 3 so as to transform it into the union oi several terms Ti, each of which is an intersection of several individual sets. Next, if in any Ti neither X appears, replace Ti by Ti ∩ (X ) and expand. Finally, group together the terms containing X and apply the distributive law 3′.)

    (c) For all sets A and B, A = B = ∅ iff A ∪ B = ∅.

    (d) The equation ( A ∩ x ) ∪ (B ) = ∅ has a solution iff B ⊆ Ā, and then any X   such that B ⊆ X ⊆Ā is a solution.

    (e)  An alternative form for solutions of the equation in part (d) is X = (B ∪ T) ∩ Ā,    where T is an arbitrary set.

    5.8. Show that for arbitrary sets A, B, C, D, and X,

    5.9.  Using the results in Exercises 5.7 and 5.8, prove that the equation

    ( A ∩ X ) ∪(B ) = ( C ∩ X) ∪ ( D )

    has a solution iff B + D . In this event determine all solutions.

    6. Relations

    In mathematics the word relation is used in the sense of relationship. The following partial sentences (or predicates) are examples of relations:

    In this section the concept of a relation will be developed within the framework of set theory. The motivation for the forthcoming definition is this: A (binary) relation is used in connection with pairs of objects considered in a definite order. Further, a relation is concerned with the existence or nonexistence of some type of bond between certain ordered pairs. We infer that a relation provides a criterion for distinguishing some ordered pairs from others in the following sense. If a list of all ordered pairs for which the relation is pertinent is available,then with each may be associated yes or no to indicate that a pair is or is not in the given relation. Clearly, the same end is achieved by listing exactly all those pairs which are in the given relation. Such a list characterizes the relation. Thus the stage is set for defining a relation as a set of ordered pairs, and this is done as soon as the notion of an ordered pair is made precise.

    Intuitively, an ordered pair is simply an entity consisting of two objects in a specified order. As the notion is used in mathematics, one relies on ordered pairs to have two properties: (i) given any two objects, x and yx, y and called the ordered pair of x and y, that is uniquely determined by x and yx, yu, v x, y u, v iff x = ,u and y = v. Now it is possible to define an object, indeed, a set, which has these properties: the ordered pair of x and y, symbolized by

    {x, y},

    is the set

    {{x},{x, y}},

    that is, the two-element set one of whose members, {x, y}, is the unordered pair involved, and the other, {x},

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