Mathematical Methods in Economics. T. S. Angell Department of Mathematical Sciences University of Delaware Newark, Delaware

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1 Mathematical Methods in Economics T. S. Angell Department of Mathematical Sciences University of Delaware Newark, Delaware c September 8, 2015

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3 Contents A Basic Set Theory 1 A.1 Introduction A.2 Specification of Sets, Equality, and Subsets A.3 The Algebra of Sets A.3.1 Unions and Intersections A.3.2 Set Differences, Complements, and DeMorgan s Laws A.4 Ordered Pairs A.5 Binary Relations and Equivalence Relations A.6 Functions or Maps A.7 Orderings on Sets i

4 ii CONTENTS

5 Appendix A Basic Set Theory A.1 Introduction We assume that most readers are familiar with the use of set notation and understand the basic operations of the algebra of sets. But a review can be helpful and there are certain particular matters that are worth emphasizing. So we devote this Appendix to a review of the basic ideas. Of course, whole books have been written on the subject and scores of textbooks contain this basic information. While these works discuss the subject with varying levels of sophistication, or point of view is the naive one. Thus we take, as basic undefined concepts, that of element, set, and the relation of belonging to. This is the point of view of the book of P. R. Halmos, Naive Set Theory [?] which is still probably the best exposition for the aspiring student. Much of what is contained in this brief appendix follows the early part of his exposition. To quote Halmos, A pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets.... An element of a set may be a wolf, a grape, or a pigeon. If we denote the set of wolves by W and a particular wolf by w, then the statement w W is the statment that w is a member of or belongs to the set W. Sometimes the words collection, or class, or family are used synonymously with the word set. Some authors reserve the word class to describe a set of sets and the word family to describe a set of classes. Thus, to continue our example, we can speak of the the set W as being an element of the class of sets of different species of mammals, and of the class of mammals 1

6 2 APPENDIX A. SETS as belonging to the family of vertibrates. Again, having pointed out this usage, we use these terms with some fluidity in our exposition. What is important, really, is clarity. There are some logical, or better, grammatical niceties, that we need to discuss before we begin. Following Halmos, we list seven logical operators which will be used throughout to construct sentences describing sets. They are and, or (in the sense of either or or both ), not, if then (or implies), if and only if, for some (or there exists ), for all. The rules of sentence formation then can be listed: (i) Put not before a sentence and enclose the result in parentheses. 1 (ii) Put and or or or if and only if between two sentences and enclose the result in parantheses. (iii) Replace the dashes in if then by sentences and enclose the result in parentheses. (iv) Replace the dash in for some or in for all by a letter, follow the result with a sentence, and enclose the whole in parentheses. The practice of enclos[ing] the result in parentheses is one that is used for clarity. Most of the time, there is NO lack of clarity if we omit the parentheses e shall seldom, if ever, use them. 1 The correct answer to the question Are you going to go to the rock concert or do something else? is Yes.

7 A.2. SPECIFICATION OF SETS, EQUALITY, AND SUBSETS 3 A.2 Specification of Sets, Equality, and Subsets A set is specified or defined when its elements are completely characterized. There are two ways to do this; one either makes an exhaustive list (without regard to order!) of all elements of the set, or one gives an explicit property or attribute that actually characterizes the elements. That doing so indeed specifies a set is often stated as an axiom of set theory. Formally: Axiom: To every set A and to every condition (equivalently sentence ) S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. The set of all students registered for a particular college course at a given moment in time is given by the traditional class list. If we look at the students whose names appear in this list, then we can define a set, F, of all named students who are female. If we call the set of students whose names appear on the class list, L, then the set F is given by {s L s is the name of a female student}. A more mathematical example is the set of points in the plane R 2 that lie on the unit circle S 1 = {(x, y) R 2 x 2 + y 2 = 1}. Notice that in both these examples a generic element is named (s in the first case and (x, y) in the second) and they are required to be elements of some universal set (L in the first case and R 2 in the second). The specification of this so-called universe of discourse makes clear what types of objects we are discussing and, in more theoretical expositions, avoids certain well-known logical difficulties as, for example, the Russell Paradox ([?]) which involves only sets which do not have themselves as elemets and the set U of all such sets. Two sets, A and B are said to be equal provided they consist of exactly the same elements. In this case we write A = B; in the contrary case we write A B. We say that a set A is a subset of B provided every element of A is also an element of B and, in this case, we write A B. Notice that this is quite different from the relation belongs to ; the relation is a subset of is, as defined, a reflexive relation in the sense that it is always true that A A. This is certainly not the case with the relation. The relation A B may also be written in the reverse order as A B. In the case that A B and A B we say that A is a proper subset of B. Note that some authors

8 4 APPENDIX A. SETS use the notation A B for the relation A B and A B in the case that A is a proper subset. We do not use that notation in this book. We will speak of relations with more specificity presently. To anticipate that discussion we point out that the relation of being a subset has certain interesting properties. One we have already mentioned, that of reflexivity. We collect the three most important properties of the relation here. Denoting the universe of discourse as U, we have (1) A A for every set A U (reflexivity) ; (2) For sets A and B, subsets of U, A B and B A implies A = B (antisymmetry) ; (3) For sets A, B, C U, A B and B C implies A C (transitivity). A relation on a class of sets, in this case all subsets of the universe U, with these three properties defines what we will call a partial order (more on partial orderings presently). In this case, we say that set class of all subsets of the set U, denoted by P(U), is partially ordered by inclusion 2. The set P(U) is called the power set of U It is important to note that in this example of a partially ordered set, the properties of reflexivity and antisymmetry can be combined into a single statement: A = B if and only if A B and B A. (A.1) This statement embodies the basic strategy for showing that two sets are equal: show every element of A is an element of B and that every element of B is also an element of A, i.e., that each set is a subset of the other. It is often convenient to have a subset of the universe of discourse, U, that contains no elements of U. This set is called the empty set and is denoted by the symbol. If we agree, as we do, that a set is specified by characterizing the properties that the elements must have, then we may use any false statement to specify the empty set. Thus, for example, we may write = {u U u u}. 2 We remark that there are other kinds of relations that can be defined on P(U). For example, equality is a relation which is certainly reflexive, and transitive; however, rather than being anti-symmetric, it is symmetric since A = B implies that B = A. Such relations on a set are called equivalence relations

9 A.3. THE ALGEBRA OF SETS 5 This set is called the empty set or, sometimes, the null set. It is a subset of any given set. In particular, in any collection of sets, there can be at most one empty set since, were there more, each would be subset of the other. A.3 The Algebra of Sets In this section we consider useful ways of combining sets to form new ones. We assume that we have a fixed universe of discourse, U, and that all sets are subsets of U. We will not always mention this universe in stating definitions. The familiar operations which we study in this section are: union, intersection, complementation, and powers. These operations are all fundamentally related to the relation of inclusion. A.3.1 Unions and Intersections We begin with the operation of set union. Given sets A and B, their union, written A B is defined by A B = {x U x A or x B}. 3 (A.2) For example, if A = {1, 5, 9, 7, 3} and B = {8, 4, 2, 6} then A B = {1, 2, 3, 4, 5, 6, 7, 8, 9}. (Note the only things here that matters are the elements themselves, not the order in which we happen to write them down!). Alternately, using the symbol N for the natural numbers (set of positive integers) we can describe the union as A B = {x N x is used in Soduku puzzles}. The operator is commutative, associative, and idempotent. That is (i) A B = B A, (ii) (A B) C = A (B C), and (iii) A A = A. Likewise it is true that A = A and A U = U or, more generally, A B if and only if A B = B. All of these statements are statements about the equality of two sets. As such, they can be proved by application of the definitions and our logical operators, although they are so elementary that few would bother to write down a proof. But every serious student should prove them once in a lifetime. Here is an example: we prove the statement Proposition A.3.1 A B if and only if A B = B. 3 REMEMBER: or means either or or both.

10 6 APPENDIX A. SETS Proof: Suppose first that A B. We then must show that A B = B. Following our basic rule (A.1) we first, check that A B B and then prove the reverse inclusion. So, if x A B then either x A or x B or both. But A B means that for every x A we must have x B so x A B implies x B. So the first inclusion is proved. Now we establish the reverse inclusion B A B. To do this, choose x B. Then, by definition of union x A or x B hence x A B. This completes the proof of sufficiency: if A B then A B = B. To prove the reverse implication, that is, to prove the necessity of the left hand statement, suppose that A B = B. We will be done if we can show that A B. To this end, choose x A. Then, x A B and since this latter set is, by hypothesis, just the set B we have x B. There are two things that are illustrated in this proof. First, the use of (A.1) to prove that two sets are equal. The other is the structure of an if and only if proof. If the reader is unsure of this reasoning, hopefully careful study of the arguments above will be helpful. The operation of intersection has many similarities with the operation of union. Given two sets A and B, their intersection A B is defined by A B = {x U x A and x B} (A.3) Notice that the definition is symmetric in A and B in the sense that A B = B A. A list of elementary properties of the intersection operator is given here: A =, A B = B A, A (B C) = (A B) C, A A = A,, A B if and only if A B = A. Notice that the last of these properties shows, along with Proposition A.3.1, that set inclusion can either be described in terms of unions or intersections. If two sets have no common elements, then they are said to be disjoint and we write A B =. In the case that we have a collection of sets, any two of which are disjoint, then we say that the collection is pairwise disjoint.

11 A.3. THE ALGEBRA OF SETS 7 We now have two operations defined on sets, union and intersection. The natural question now is to ask how these two operations are related to each other. This question is answered by the distributive laws: A (B C) = (A B) (A C) (A.4) A (B C) = (A B) (A C. (A.5) As an exercise intended to illustrate, once more, a set-theoretic argument, let us prove the first of these distributive laws. Proposition A.3.2 For any sets A, B and C subsets of the set U, A (B C) = (A B) (A C). Proof: If x belongs to the left-hand side of this equation, then x A and x B or x C or both. If x B then x A B and so is an element of the set on the right. Likewise, if x C then x A C and so, again, x belongs to the right hand side. This shows that the set on the right-hand side includes the set on the left. To prove the reverse inclusion, suppose x belongs to the set on the right. Then either x A B or x A C or both. If x A B then x A and x B. So x B C and hence x A (B C). Likewise, if x A C then x C and thus x B C. So in this case as well, x A (B C) and x belongs to the left-hand set. Hence the reverse inclusion is satisfied. wwe conclude that the sets on each side of th equality are the same set. A.3.2 Set Differences, Complements, and DeMorgan s Laws The set theoretic difference A\B (also written A B) is defined by A\B = {a A a B}. In many situations we are only interested in subsets of a given set X (the universe of discourse). The complement A c of a set A with respect to X is defined by A c = X\A = {a X a A}.

12 8 APPENDIX A. SETS We can now formulate and prove De Morgan s Laws. These are rules that relate complements of unions to intersections of complements, and complements of intersections to unions of complements. It is surprising how useful they are and how often they are used 4. In the case of just two sets A, B X, these rules are simple to write down and understand in terms of Venn diagrams and the reader is invited to do so. In this simple case they read and X \ (A B) = (X \ A) (X \ B), or (A B) c = A c B c. X \ (A B) = (X \ A) (X \ B), or (A B) c = A c B c. (A.6) (A.7) Proposition A.3.3 Assume that A 1, A 2,..., A n are subsets of the set X. Then (A 1 A 2... A n ) c = A c 1 A c 2... A c n, and (A 1 A 2... A n ) c = A c 1 A c 2... A c n. Proof: For the first part, assume that x (A 1 A 2... A n ) c. Then x A 1 A 2... A n, and hence x A i for any i = 1, 2,..., n. This means that x A c i for all i and so x A c 1 A c 2... A c n. So we have shown that (A 1 A 2... A n ) c A c 1 A c 2... A c n. To prove the reverse inclusion, assume that x A c 1 A c 2... A c n. This means that x A c i for all i. So x A i for all i = 1, 2,..., n. It follows that x A 1 A 2... A n which means that x (A 1 A 2... A n ) c. This completes the proof of the first equality. We leave the (analogous) proof of the second equation as an exercise. De Morgan s Laws have extension to arbitrary families of sets. We first extend the notions of union and intersection to families of sets in the following way: If A is a nonempty family of sets, we define 4 DeMorgan s Laws are frequently used in programming, in particular in the construction of sorting algorithms. From the point of view of logic, they allow the substitution of equivalent statements, e.g., not S or not T being equivalent to not both S and T.

13 A.3. THE ALGEBRA OF SETS 9 and A = {a U a belongs to at least one set A A} A A A = {a U a belongs to all sets A A}. A A The distributive laws and the Laws of De Morgan extend to this case in the obvious ways, e.g., ( ) c A = A c. A A A A Families are often given as indexed sets. This means we have one basic set I (the index set and the family consists of one set A i for each element i I. We then write the family as A = {A i i I}. We may then write A i and i I i I for unions and intersections. In this setting De Morgan s Laws become A i ( i I A i)c = i I A c i and ( i I A i)c = i I A c i. Let us finish the section with a simple example. Example A.3.4 for each rational number q Q we can take the set C q := {(x, y) R 2 x 2 + y 2 = q 2 } which is just the circle with rational radius q, centered at the origin. Then we can consider C(Q) = {C q q Q}. The set C(Q) is just the family of all circles in the plane R 2 with center at the origin and rational radius. Note that this family can be thought of as a family of sets indexed by the rationals.

14 10 APPENDIX A. SETS A.4 Ordered Pairs In analytic geometry and elementary calculus it is common to introduce the coordinate, or (x, y)-plane. The horizontal axis or axis of abscissae is associated with the value of x and the vertical axis or axis of ordinates is associated with the y coordinate. The agreement that the abscissa be listed first and the ordinate next in a certain sense gives a geometric definition of the ordered pair (x, y). Likewise this construction gives a concrete example of what is called the Cartesian product 5 of two sets, in this case the two sets are two copies of the real line. In the theory of sets, we need a much more precise description of the notion of ordered pair but its introduction makes things get very technical very quickly. We have given here a quick summary for the sake of completeness. The definition given here has the disadvantage of strangeness, but the decided advantage of settling the problem of what we mean by a first element of an ordered pair. If A = {a, b} and, in the desired order a comes first then we need a careful definition. Here is the definition that we shall adopt. Definition A.4.1 The ordered pair of a and b with first coordinate a and second coordinate b, is the set (a, b) defined by (a, b) = {{a}, {a, b}}. The definition clearly specifies the first element ; it is the element that occurs in the singleton set {a}. There are some technical difficulties that need to be addressed which arise from the fact that the ordered pair is definied as a set of sets. Halmos ([?], pp ) deals with all of them. What is important is the statement that, given sets A and B, there exists a set that contains all the ordered pairs (a, b) with a A and b B. This set is called the Cartesian product of A and B, is written A B, and is characterized by the fact that A B = {x P[P(A B)] x = (a, b) for some a A and some b B}. Remark: Note that we have written x P[P(A B) so (a, b) is a set of subsets of A B. If follows that one ordered pair is an element of P(A B), while a set of ordered pairs is then a set of subsets of A B i.e., an element of P[P(A B)]. 5 After all, it was René Desartes who introduced numerical components into geometry.

15 A.5. BINARY RELATIONS AND EQUIVALENCE RELATIONS 11 If R A B then the sets R A = {a A for some b B, (a, b) R} and R B = {b B for some a A, (a, b) R} are called the projections of R onto the first and second coordinates respectively. Having seen the rigorous definition, we will henceforth treat ordered pairs less formally as is usually done. Again, there are several facts that can be easily checked and we leave them as exercises: Exercise A.4.2 If A, B, X, and Y are sets, then (a) If either A = or B =, then A B =. The converse is also true. (b) A X and B Y implies A B X Y. If A B then A B X Y implies A X and B Y. (c) The following distributive laws hold: (i) (A B) X = (A X) (B X) ; (ii) (A B) (X Y ) = (A X) (B Y ) ; (iii) (A B) X = (A X) (B X). A.5 Binary Relations and Equivalence Relations We start with two sets A and B. Then a binary relation R on a set A B is a proposition such that, for every ordered pair (a, b) A B, one can decide if a is related to b or not. It is simply a restricted set of ordered pairs. Formally, Definition A.5.1 A binary relation in a set A B is a subset R A B. The statement (a, b) R is written as a R b.

16 12 APPENDIX A. SETS Example A.5.2 (a) For any set A A 6 the diagonal = {(a, a) a A} is the relation of equality. The relation [(A A) \ ] is the relation of inequality. (b) The relation between two real numbers is the set {(x, y) R R x coincides or lies to the left of y } R R. (c) In P(A) the relation of set inclusion, B A, is given by {(A, B) P(A) P(A) every element of B is an element of A }. (d) For any set X, let R be the relation on X P(X) defined by (x, A) R if and only if x A. This is the relation of membership in a set. If A B is a set with a binary relation R and C A, D B then the relation R (C D) is a binary relation on the set C D. It is called the relation induced by R on C D. Of all the relations, one of the most important is the equivalence relation. We will denote such a relation by the symbol and write a b when we mean that a is equivalent to b. We will also say that a is similar to b. Definition A.5.3 An equivalence relation on a set X is a binary relation on X which is reflexive, symmetric and transitive, i.e. (a) for all a X : a a (reflexive). (b) a b implies b a (symmetric). (c) a b and b c implies a c (transitive). We begin with some simple examples. Example A.5.4 (a) The relation is an equivalence relation. (b) In N the relation {(x, y) N N x y is divisible by 2} is an equivalence relation. 6 We will often say that a relation is defined on A to mean a relation on A A, a slight abuse of language that should cause no problem.

17 A.5. BINARY RELATIONS AND EQUIVALENCE RELATIONS 13 (c) Let f : X Y be a function. equivalence relation on X. Then {(x 1, x 2 ) X X f(x 1 ) = f(x 2 )} is an (d) Let T be the set of all triangles in the plane R 2. Then the relation of congruence, familiar from elementary Euclidean geometry is an equivalence relation. Let us check the assertion (b). First, reflexivity. For alll x we have x x = 0 and 0 is divisible by 2. Hence the relation is reflexive. Moreover, since y x = ( 1) (x y) it is clear that if 2 (x y) then 2 (y x). Hence the relation is symmetric. Finally, if 2 (x y) and 2 (y z), then since x z = (x y) + (y z), it is clear that 2 (x z). So the relation is also transitive and hence is an equivalence relation. Suppose that is an equivalence relation on the set X. If x X let E(x, ) denote the set of all elements y X such that x y. The set E(x, ) is called the equivalence class of x for the equivalence relation. Since is an equivalence relation, the equivalence classes have the following properties: 1. Each E(x; ) is non-empty for, since x x, x E(x; ). 2. Let x and y be elements of X. Since is symmetric, y E(x; ) if and only if x E(y; ). 3. If x, y X the equivalence classes E(x; ) and E(y; ) are either identical or they have no members in common. Indeed, suppose, first, that x y. Let z E(x; ). Then, by symmetry, since z x we have also x z. Hence, by transitivity, z y and so, by symmetry, y z. This shows that E(x; ) E(y; ). By the symmetry of we see that E(y; ) E(x; ). Hence E(x; ) = E(y; ). Finally, notice that if the points x, y X are not related then E(x; ) E(y; ) =. Indeed, if z E(x; ) E(y; ) then x z and y z and so x z and z y. Therefore x y which is a contradiction. These facts lead to the following assertions concerning the family, F, of equivalence classes of the equivalence relation : 1. Every element of the family F is non-empty.

18 14 APPENDIX A. SETS 2. Each element x X belongs to one and only one of the sets in the family F. 3. x y if and only if x and y belong to the same set in the family F. Otherwise said, an equivalence relation subdivides a set (or partitions the set) into the union of a family of non-overlapping, non-empty subsets. Since, in most discussions, there is only one equivalence relation that is relevant, we will often write simply E(x) instead of E(x; ) is no confusion can arise. Here is an example which is perhaps the first most students see when they discuss number systems. Example A.5.5 In the construction of the rational numbers, which we will denote by Q, we first introduce ratios of integers p/q where p N and q N. If p/q represents a point on the number line, then the ratios kp/kq must represent the same point and hence the same rational number. Thus, two ratios p/q and r/s represent the same rational number and can be treated as equal and can be substituted for one another in proofs involving rational numbers whenever the equality ps = rq is true. Now, let us define a relation on N N by (p, q) (r, s) if and only if ps = rq. We check that this is an equivalence relation as follows: (a) (Reflexivity): pq = pq hence (p, q) (p, q). (b) (Symmetry): If ps = rq then rq = ps and so (p, q) (r, s) implies (r, s) (p, q). (c) (Transitivity): If ps = rq and rt = vs, then (pt) s = (ps) t = (rq) t = (rt) q = (vs) q = (vq) s and thus pt = vq since s 0. Hence (p, q) (r, s) and (r, s) (v, t) implies (p, q) (v, t).

19 A.5. BINARY RELATIONS AND EQUIVALENCE RELATIONS 15 This argument shows that the rational numbers can be viewed as equivalence classes of ratios of integers modulo the relation given in the example. As a final example consider the following: Example A.5.6 : Consider the set Z and let n be a fixed positive integer. Define a relation n by x n y provided (x y) is divisible by n. This relation is called the relation of congruence modulo n. It is easy to check that this is an equivalence relation on Z. (See the special case for n = 2 treated above.) Moreover, there are n equivalence classes. Each integer x is uniquely expressible in the form x = q n + r, where q and r are integers and 0 r n 1. (The integers q and r are called the quotient and the remainder respectively. ) Hence each x is congruent modulo n to one of the n integers 0, 1,..., n 1. The equivalence classes are E 0 = {..., 2n, n, 0, n, 2n,...} E 1 = {..., 1 2n, 1 n, 1 + n, 1 + 2n,...}.. E n 1 = {..., n 1 2n, n 1 n, n 1, n 1 + n, n 1 + 2n,...} Formallly, the domain of a relation, R, on X Y is the set of all first coordinates of the members of R while, in this context, the range is the set of all second coordinates. Formally while dom (R) = {x X for some y Y, (x, y) R}, rng (R) = {y Y for some x X, (x, y) R}. The inverse of a relation R, denoted R 1, is obtained by reversing each of the pairs belonging to R. Thus R 1 = {(y, x) Y X (x, y) R}.

20 16 APPENDIX A. SETS Hence the domain of the inverse is the range of R and the range of R 1 is always the domain of R. If R and S are relations, then the composition R S is defined as {(x, z) X Z for some y, (x, y) S and (y, z) R}. Example A.5.7 If R = {(1, 2)} and S = {(0, 1)} then R S = {(0, 1)} while S R =. Concerning compositions and inverses we have the following result Proposition A.5.8 Let R, S, and T be relations. Then (a) (R 1 ) 1 = R. (b) (R S) 1 = S 1 R 1. (c) R (S T ) = (R S) T Proof: (of (b)) We have (x, a) (R S) 1 (x, z) R S for some y, (x, y) S and (y, z) R. Consequently, (z, x) (R S) 1 if and only if (y, z) R 1 and (y, a) S 1 for some y. But this is the condition that (z, x) S 1 R 1. A.6 Functions or Maps We now define the idea of a function (or a mapping) in terms of sets. This is not so unusual since we often think of a function in terms of its graph which consists of a set of ordered pairs: given two sets X and Y a function is determined provided we specify a set of ordered pairs (the graph of the function) in X Y with the additional property that no two distinct pairs have the same first element. Hence a function is a particular example of a relation! Not every subset of ordered pairs will do however; to be a function the ordered pairs must satisfy a particular condition.

21 A.6. FUNCTIONS OR MAPS 17 Definition A.6.1 Let X and Y be two sets. A map f : X Y (or a function with domain X and range Y ) is a subset f X Y with the property: for each x X, there is one, and only one, y Y satisfying (x, y) f. It is usual to write y = f(x) instead of (x, y) f and say that y is the value f assumes at x, or that y is the image of x under f, or that f sends x to y. The usual way to define a map is to specify its domain X and the value of the function at each x X. We often write x f(x). Here are some examples. Example A.6.2 : (a) Suppose k Y is fixed. Then the map defined for all x X by x k is called a constant map. Note that a map need not send distinct points of X to distinct points of Y, nor do we require it to take on all values in its range. (b) The map x x of X onto itself is called the identity map on X. We will often write this as Id X. (c) If A X the map i : A X given by a a is called the inclusion map of A into X. (d) For any sets X, Y the map p 1 : X Y X determined by (x, y) x is called the projection onto the first coordinate. Similarly p 2 : X Y Y given by (x, y) y is called the projection onto the second coordinate. We have stated above that it is not required that every point in the range be a value that is taken on by a given function. That is the motivation for the following definition. Definition A.6.3 Let f : X Y. Then (1) For each A X, f(a) = {f(x) Y x A} Y is called the image of A in Y under f. (2) For each B Y, f 1 (B) = {x X f(x) B} is called the inverse image of B in X under f. Again, it is a good idea to see some examples.

22 18 APPENDIX A. SETS Example A.6.4 (a) Let X = [ 1, 1], Y = [0, 2] and f : X Y be given by x x 2. Then f 1 ({1/4}) = { 1/2, 1/2}. This shows tat the inverse image of a single point may well be a set in the domain. This cannot happen, of course, if f is one-to-one (see definition below). 7 (b) Let X = [ 1, 1], Y = [0, 2], and f : X Y be x x 2. Then f[0, 1] = [0, 1] and 2 4 f 1 [0, 1] = [ 1, 1] (c) Let f : X Y. If p 1, p 2 are the projections defined in the preceeding example (A.6.2, part (d)), we have f(a) = p 2 [f (A Y )] and f 1 (B) = p 1 [f (X B)]. It is useful to think, explicitly, about how a function f : X Y induces a map from P(X) P(Y ). This induced map is defined by A f(a) we call this induced map f as well. Likewise, f : X Y also induces a map f 1 : P(Y ) P(X) by B f 1 (B) called the inverse map. Of these two maps, the most well-behaved is the inverse map f 1 and, in some sense, it is the most important. Proposition A.6.5 : Let f : X Y. then the inverse map f 1 : P(Y ) P(X) preserves union and intersection. Precisely (a) f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). (b) f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). Proof: We leave (a) as an exercise and prove (b). x f 1 (B 1 B 2 ) if and only if f(x) (B 1 B 2 ) if and only if f(x) B 1 and f(x) B 2 if and only if x f 1 (B 1 ) and x f 1 (B 2 ) if and only if x [f 1 (B 1 ) f 1 (B 2 )]. As a corollary, we can restate the result for arbitrary intersections; the proof is analogous. 7 While we usually make a careful distinction between a singleton set {x} X and a point x X we often abuse notation and write simply f 1 (y) instead of f 1 ({y}).

23 A.6. FUNCTIONS OR MAPS 19 Corollary A.6.6 Let f : X Y and let f 1 be the inverse map. Then if B be a family of subsets of the set Y we have f 1 ( B B B ) = B B f 1 (B) and f 1 ( B B B ) = B B f 1 (B). We have said that f 1 is better behaved because the last result is not true for the induced map f. Indeed we have the counterexample: Example A.6.7 : Let f : R R be the constant map x 1. B = [2, 3]. Then A B = and so Let A = [0, 1] and = f(a B) f(a) f(b) = {1}. We do find, however, that f preserves unions as the next result shows. Proposition A.6.8 let A be a family of subsets of the set X. If f : X Y, then for the induced map f : P(X) P(Y ) we have ( ) f A = ( ) f(a), and f A f(a). A A A A A A A A We leave the proof of this last result to the reader. Note that, in general, we do not have equality in the last case as the above example shows. To further clarify matters let us look at another example. Example A.6.9 Let X = {x 1, x 2 } and Y = {y}. Define f : X Y by f(x 1 ) = f(x 2 ) = y, and let A 1 = {x 1 }, A 2 = {x 2 }. Then A 1 A 2 = and consequently f(a 1 A 2 ) =. On the other hand, f(a 1 ) = f(a 2 ) = {y} and so f(a 1 ) f(a 2 ) = {y}. This means that f(a 1 A 2 ) f(a 1 ) f(a 2 ). The problem here stems from the fact that y belongs to both f(a 1 ) and f(a 2 ) but only as the image of two different elements x 1 A 1 or x 2 A 2 ; there is no common element x A 1 A 2 which is mapped into y. This cannot happen if f is one-to-one. As for compositions, we have the usual result familiar from calculus.

24 20 APPENDIX A. SETS Proposition A.6.10 Let f : X Y and g : Y Z. Then (g f) 1 = f 1 g 1. Proof: x (g f) 1 (C) if and only if g f(x) C if and only if f(x) g 1 (C) if and only if x f 1 [g 1 (C)] if and only if x f 1 g 1 (C). If f : X Y takes on every value in its range, f is called surjective (or a surjection or onto). Note that, for a surjective f we have for all B Y, f[f 1 (B)] = B. If f sends distinct elements of X to distinct elements of Y, then f is call injective (or and injection or one-to-one). Otherwise said, f in injective provided that x 1 x 2 iimplies f(x 1 ) f(x 2 ). This is equivalent to the statement that f(x 1 ) = f(x 2 ) if and only if x 1 = x 2. A function that is both injective and surjective is called bijective or a bijection. Note that f is a bijection if and only if for all y Y, f 1 ({y}) is a single point. In this case (f 1 ) 1 = f. Example A.6.11 Consider the mapping f : R R defined by x 2 x + 3. Then f is certainly a bijection with inverse mapping y 1 2 y 3 2. Indeed 1 2 (2x + 3) 3 2 = x. Finally, we return to the notion of an indexed family of sets that we met in Section A3. Let I be any set and F a family of subsets of a universal set U. Suppose, moreover, that f : I F. Then we write f(i) = A i. In this way the function I is said to index the sets {A i F f(i) = A i, i I}. If the map f is surjective, then we say that the family F has been indexed by I. In particuar, if I = N then, to specify such a function simply defines what we usually mean by a sequence of sets, and we write {A 1, A 2,...}. In the case that f is surjective, we say that F is a countable family of subsets of U. Example A.6.12 In R n, let B(0) be the set of all sets of the form {x R n n x 2 i < r 2, r R}. These sets are the points in R n whose distance from the origin is less than r. Denote these sets by B r (0). Then the sets B n (0) form a countable subset of B(0). i=1

25 A.7. ORDERINGS ON SETS 21 A.7 Orderings on Sets We will frequently meet certain binary relations on various sets which are called orderings of which there are several types. Such relations are used in economics, for example, to describe preferences of various agents. Thus suppose that an n-vector x = (x 1, x 2,..., x n ) R n represents a bundle of goods available to consumers, x i representing the amount of good i in the bundle. Thus, for example, if the first component represents the number of refrigerators measured in units while the second component represents wheat measured in bushels then (2, 3.659,...) R n is a bundle of goods consisting, among other things, of two refrigerators and bushels of wheat. In describing consumer behavior, we generally make the assumption that one and only one of the following alternatives holds: 1. a bundle x is preferred to the bundle y; 2. the consumer is indifferent in the choice of two bundles; 3. the bundle y is preferred to the bundle x. Note that these alternatives, taken together, imply that the consumer can decide unambiguously between two bundles. This situation suggests that we introduce some kind of structure that reflects preference, i.e., some way of ordering bundles to reflect consumer desires. Definition A.7.1 A binary relation R in a set A is said to be a preorder on A if it is reflexive and transitive, i.e., (a) for all a A, a R a. (b) If a R b and b R c then a R c. A set, together with a definite preorder, is called a preordered set. It is traditional to write a preorder with the symbol or with. Thus a preceeds b, or b is preceeded by a, or, in economics, b is preferred to a is written a b. The symbol (A, ) denotes a preordered set. Notice that if B A and if A is preordered by then, by default, this preorder induces a preorder on B.

26 22 APPENDIX A. SETS Example A.7.2 (a) In any set, the relation (the diagonal relation) on a set A A is a preorder and a b means a = b. Note that we no not assume in the definition that any two elements can be compared. In other words, we do not require that either a b or b a for all a, b A. (b) In the set R the relation {(x, y R R x y} is a preorder. On the other hand {(x, y) R R x < y} is not a preorder. (Why?) (c) (IMPORTANT!) For any set X, consider the power set P(X). The relation A B defined by A B if and only if A B is a preordering of P(X). In this particular case, we say that P(X) is preordered by inclusion. By putting other conditions on a preordering, different types of orders can be obtained. Definition A.7.3 If a preordering on A satisfies the additional property of antisymmetry, i.e., a b and b a a = b, then it is called a partial ordering. In this case A is called a partially ordered set. A partially ordered set that is also a chain (see Definition 1.3 a) is called a totally ordered set. The set P(X) is partially ordered by inclusion since the preorder is also antisymmetric. In general, the set P(X) is not a chain. In this context, what does a chain look like? One example is the following: for each n N, let A n P(X) and suppose that, for each n, A n+1 A n and A n A n+1. Then the set of subsets {A n } n=1 constitutes a chain in P(X) with respect to the partial ordering of inclusion. Example A.7.4 Let X = [ 1, 1] R and let A n = [ 1, 1 ], n = 1, 2,. Then this set n n forms a chain, namely [ 1, 1] [ 1 2, 1 2 ] [ 1 3, 1 3 ]

27 A.7. ORDERINGS ON SETS 23 As another, and important, example we consider the following. Example A.7.5 Consider the set R n. Let x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ). Then we can introduce a partial order in R n by x y x i y i for all i = 1, 2,..., n. Here is the usual ordering on the real line. Note that this is certainly a reflexive, transitive, and anti-symmetric relation so that is indeed a partial ordering of R n. Note further that not every two elements can be compared. Thus, for example, in R 2, the vectors (1, 2) and (2, 1) are not comparable. This last example contrasts with the usual ordering on the real line, R, where every element can be compared. This leads to an important special case of a partial order. Definition A.7.6 Let A be a set. A total or linear order on the set A is a partial order such that, for all x, y A, x y either x y or y x whenever x and y are both in the domain and range of the order relation. The usual order on the real line is the obvious example. We remark that, in this terminology, a chain in a partially ordered set is a totally ordered family. Here is another example which is important in a number of applications including integer programming algorithms and sorting algorithms in computer sciennce. Example A.7.7 (Lexicographical Order) Let X be the set of all infinite sequences of real numbers. Define a relation L on X by a L b provided, for the smallest integer i o such that a io b io, a io < b io. This order is called lexicographical order since it is the same kind of order used in common dictionaries. In fact, this order is a total, or linear, order. Indeed, it is clearly reflexive. To check transitivity, note that if a L b and b L c then for minimial i o and j o, we have a io < b io and b jo < c jo. If i o < j o then a io < b io = c io and so a L c. On the other hand, if j o < i o then a jo = b jo < c jo and so, again, a L c. The property of anti-symmetry is easy to check. Finally, since any two sequences can be compared, this partial ordering is indeed a total order.

28 24 APPENDIX A. SETS Let us finish this section by discussing some standard ideas pertaining to partially ordered sets that will be of some use in discussions of some of the ideas of Pareto and multi-criteria optimization. Definition A.7.8 Let (A, ) be a partially ordered set with partial order given by. Then (a) m A is called a maximal element in A provided m a implies m = a. The set A is said to have a greatest element m provided, for all a A, a m. (b) a o A is called an upper bound for a subset B A provided, for all b B, b a o. (c) B A is called a chain in A if each two elements in B are related. It is important to distinguish between maximal and greatest elements. Example A.7.9 (a) Consider the set consisting of the union of the sets A = {2, 4, 6} and B = {3, 9, 27}. Partially order the union with the relation a b provided a is a factor of b. Then there is no largest element, but both 6 and 27 are maximal. (b) If we take the union of A = {2, 4, 6} and B = {1, 3, 9, 27}, then 1 is both a minimal element and a least element. We end this appendix with some remarks on functions that preserve order when the domain and range of the function are ordered in some way. This is an important question in Economics where it arises in the context of the existence of a scalar-valued utility function. We confine ourselves to some elementary remarks. In what follows, we are given two partially ordered sets {X, } and {Y, <}. Definition A.7.10 A function F : X Y is said to be order preserving or sometimes isotone relative to the orders on X and Y provided f(u) < f(v) or f(u) = f(v) whenever u, v X are such that u v. The situation of interest to us is the case in which is a partial order, while < is a linear, i.e., a complete order, in fact, Y = R and the order is the usual order on R. Unfortunately, some extra conditions must be put on the sets before an isotone mapping will exist even in the case that the order relation is a total order.

29 A.7. ORDERINGS ON SETS 25 We can see that this is the case by showing that lexicographic order does not admit an order-preserving or isotone map into R with its usual order. The proof uses two elementary facts about the system of real numbers, namely that any interval of positive length contains a rational number, and that the cardinality of the set of rationals is less than the cardinality of the set of reals. In particular, there cannot be an injective map of R > into the rational numbers Q. The argument is by contradiction. Proof: Suppose we are given a map u : R 2 > R that represents lexicographic order in R 2. Were this the case, then, given any x R >, u(x, 0) < u(x, 1) since (x, 0) (x, 1) the first components being the same and 0 < 1. Thus we can define an interval I(x) = [u(x, 0), u(x, 1)] R, of positive length. Now, let x, y R > with x y. Without loss of generality, we may assume that y < x. Then (y, 1) (x, 0) in the lexicographic order. It follows that I(x) I(y) = for (y, 0) (y, 1) (x, 0) (x, 1). Now let I = I(x) and define ϕ : R > I by x R > ϕ(x) = I(x). This map is injective since I(x) I(y) =. Now it is a property of R that every interval contains a rational number. To finish the proof, we pick a function that chooses one rational from each of the intervals 8. Call this map ψ. Then ψ : I Q and ψ(i(x)) is, for each x, some rational number in the interval I(x). Since the {I(x)} x R> is a disjoint family, the map ψ is injective and hence ψ ϕ : R > Q is an injective map. Hence card (R > ) card (Q) which is false. 8 Here, we are using the Axiom of Choice which we take as a fundamental axiom of Set Theory.

30 26 APPENDIX A. SETS

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32 28 BIBLIOGRAPHY [12] P. A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge MA, [13] P. A. Samuelson, Economics: an introductory analysis, Fifth Ed., McGraw-Hill, New York, NY, [14] A. Takayama, Mathematical Economics, Second Ed., Cambridge University Press, Cambridge, UK, 1985.