An Introduction to Mathenatical analysis in econonics Dean Corbae and Juraj zeman December 2002 IStill Preliminary. not to be photocopied or distributed without permission of the authors
An Introduction to Mathematical Analysis in Economics1 Dean Corbae and Juraj Zeman December 2002 1Still Preliminary. Not to be photocopied or distributed without permission of the authors
Contents 1 Introduction 1.1 Rules of logic 1.2 Taxonomy of Proofs 1.3 Bibliography for Chapter 1 2 Set Theory 2.1 Set Operations 2.1.1 Algebraic properties of set operations 2.2 Cartesian Products 2.3 Relations 2.3. 1 Equivalence relations 2.3.2 Order relations 2.4 Correspondences and Functions 2.4.1 Restrictions and extension 2.4.2 Composition of functions 2.4.3 Injections and inverses 2.4.4 Surjections and bijections 2.5 Finite and Infinite Sets 2.6 Algebras of Sets 2.7 Bibliography for Chapter 2 2.8 End of Chapter Problems 3 The Space of Real Numbers 3.1 The Field Axioms 46 3.2 The Order Axioms 3.3 The Completeness Axiom 3.4 Open and Closed Sets 3.5 Borel sets
Contents 1 Introduction 13 1.1 Rules of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Taxonomy of Proofs . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Bibliography for Chapter 1 . . . . . . . . . . . . . . . . . . . . 19 2 Set Theory 21 2.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Algebraic properties of set operations . . . . . . . . . . 24 2.2 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Equivalence relations . . . . . . . . . . . . . . . . . . . 25 2.3.2 Order relations . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Correspondences and Functions . . . . . . . . . . . . . . . . . 30 2.4.1 Restrictions and extensions . . . . . . . . . . . . . . . 32 2.4.2 Composition of functions . . . . . . . . . . . . . . . . . 32 2.4.3 Injections and inverses . . . . . . . . . . . . . . . . . . 33 2.4.4 Surjections and bijections . . . . . . . . . . . . . . . . 33 2.5 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Algebras of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 Bibliography for Chapter 2 . . . . . . . . . . . . . . . . . . . . 43 2.8 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 44 3 The Space of Real Numbers 45 3.1 The Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The Order Axioms . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 The Completeness Axiom . . . . . . . . . . . . . . . . . . . . 50 3.4 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3
CONTENTS Bibilography for Chapter 3 End of Chapter Problems 4 Metric Spaces 4.1 Convergence 4.1.1Co nce of functions 4.2 Completeness 4.2.1 Completion of a metric space 4.3 Compactness 4.4 Connectedness 4.5 Normed Vector Spaces 4.5.1 Convex sets 4.5.2 A finite dimensional vector space: Rn 的70888299 4.5.3 Series 4.5.4 An infinite dimensional vector space: ep 4.6 Continuous Functions 105 4.6.1 Intermediate value theorem 108 4.6.2 Extreme value theorem 110 4.6.3 Uniform continuity 4.7 Hemicontinuous Correspondences 4.7.1 Theorem of the maximum 4.8 Fixed Points and Contraction Mappings 4.8.1 Fixed points of functions 4.8.2 Contractions 4.8.3 Fixed points of correspondences 4.9 Appendix- Proofs in Chapter 4 138 4.10 Bibilography for Chapter 4 144 4.11 End of Chapter Problems 145 5 Measure Spaces 14 5.1.1 Outer measure 5.1.2 -measurable sets 154 5. 1.3 Lebesgue meets borel 5.1.4 L-measurable mappings 159 5.2 Lebesgue Integration 170 5.2.1 Riemann integrals 170 5.2.2 Lebesgue integrals
4 CONTENTS 3.6 Bibilography for Chapter 3 . . . . . . . . . . . . . . . . . . . . 63 3.7 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 64 4 Metric Spaces 65 4.1 Convergence ..................... . . . . . . . 68 4.1.1 Convergence of functions . . . . . . . . . . . . . . . . . 75 4.2 Completeness .................... . . . . . . . 77 4.2.1 Completion of a metric space. . . . . . . . . . . . . . . 80 4.3 Compactness .................... . . . . . . . 82 4.4 Connectedness .................... . . . . . . . 87 4.5 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 88 4.5.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.2 A finite dimensional vector space: Rn . . . . . . . . . . 93 4.5.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5.4 An infinite dimensional vector space: !p . . . . . . . . . 99 4.6 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . 105 4.6.1 Intermediate value theorem . . . . . . . . . . . . . . . 108 4.6.2 Extreme value theorem . . . . . . . . . . . . . . . . . . 110 4.6.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . 111 4.7 Hemicontinuous Correspondences . . . . . . . . . . . . . . . . 113 4.7.1 Theorem of the Maximum . . . . . . . . . . . . . . . . 122 4.8 Fixed Points and Contraction Mappings . . . . . . . . . . . . 127 4.8.1 Fixed points of functions . . . . . . . . . . . . . . . . . 127 4.8.2 Contractions . . . . . . . . . . . . . . . . . . . . . . . . 130 4.8.3 Fixed points of correspondences . . . . . . . . . . . . . 132 4.9 Appendix - Proofs in Chapter 4 . . . . . . . . . . . . . . . . . 138 4.10 Bibilography for Chapter 4 . . . . . . . . . . . . . . . . . . . . 144 4.11 End of Chapter Problems . . . . . . . . . . . . . . . . . . . . 145 5 Measure Spaces 149 5.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1.1 Outer measure . . . . . . . . . . . . . . . . . . . . . . 151 5.1.2 L−measurable sets . . . . . . . . . . . . . . . . . . . . 154 5.1.3 Lebesgue meets borel . . . . . . . . . . . . . . . . . . . 158 5.1.4 L-measurable mappings . . . . . . . . . . . . . . . . . 159 5.2 Lebesgue Integration . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.1 Riemann integrals . . . . . . . . . . . . . . . . . . . . . 170 5.2.2 Lebesgue integrals . . . . . . . . . . . . . . . . . . . . 172
CONTENTS 5.3 General Measure 5.3.1 Signed measures 5. 4 Examples Using Measure Theory 5.4.1 Probability Spaces 5.4.21 195 5.5 Appendix- Proofs in Chapter 5 200 5.6 Bibilography for Chapter 5 6 Function Spaces 213 6. 1 The set of bounded continuous functions 6.1.1 Completeness 216 6. 1.2 Compactness 6. 1.3 Approximation arability of C(X) 6. 1.5 Fixed point theorems 6.2 Classical Banach spaces: L 229 6.2.1 Additional Topics in Lp(X 235 6.2.2 Hilbert Spaces(L2(X)) 37 6.3 Linear operators 241 6.4 Linear functionals 6.4.1 Dual spaces 248 6.4.2 Second Dual Space 6.5 Separation Results 6.5.1 Existence of equilibrium 260 6.6 timization of Nonlinear Operators 262 6.6.1 Variational methods on infinite dimensional vector spaces 262 6.6.2 Dynamic Programming 274 6.7 Appendix- Proofs for Chapter 6 284 6.8 Bibilography for Chapter 6 297 7 Topological Spaces 7.1 Continuous Functions and Homeomorphisms 7.2 Separation Axioms 7.3 Convergence and Completeness 305
CONTENTS 5 5.3 General Measure . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3.1 Signed Measures . . . . . . . . . . . . . . . . . . . . . 185 5.4 Examples Using Measure Theory . . . . . . . . . . . . . . . . 194 5.4.1 Probability Spaces . . . . . . . . . . . . . . . . . . . . 194 5.4.2 L1 .................... . . . . . . . . . 195 5.5 Appendix - Proofs in Chapter 5 . . . . . . . . . . . . . . . . . 200 5.6 Bibilography for Chapter 5 . . . . . . . . . . . . . . . . . . . . 211 6 Function Spaces 213 6.1 The set of bounded continuous functions . . . . . . . . . . . . 216 6.1.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . 216 6.1.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . 218 6.1.3 Approximation . . . . . . . . . . . . . . . . . . . . . . 221 6.1.4 Separability of C(X) . . . . . . . . . . . . . . . . . . . 227 6.1.5 Fixed point theorems . . . . . . . . . . . . . . . . . . . 227 6.2 Classical Banach spaces: Lp . . . . . . . . . . . . . . . . . . . 229 6.2.1 Additional Topics in Lp(X) . . . . . . . . . . . . . . . 235 6.2.2 Hilbert Spaces (L2(X)) . . . . . . . . . . . . . . . . . . 237 6.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.4 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 245 6.4.1 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . 248 6.4.2 Second Dual Space . . . . . . . . . . . . . . . . . . . . 252 6.5 Separation Results . . . . . . . . . . . . . . . . . . . . . . . . 254 6.5.1 Existence of equilibrium . . . . . . . . . . . . . . . . . 260 6.6 Optimization of Nonlinear Operators . . . . . . . . . . . . . . 262 6.6.1 Variational methods on infinite dimensional vector spaces262 6.6.2 Dynamic Programming . . . . . . . . . . . . . . . . . . 274 6.7 Appendix - Proofs for Chapter 6 . . . . . . . . . . . . . . . . . 284 6.8 Bibilography for Chapter 6 . . . . . . . . . . . . . . . . . . . . 297 7 Topological Spaces 299 7.1 Continuous Functions and Homeomorphisms . . . . . . . . . . 302 7.2 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 303 7.3 Convergence and Completeness . . . . . . . . . . . . . . . . . 305
Acknowledgements To my family: those who put up with me in the past -Jo and Phil-and especially those who put up with me in the present- Margaret, Bethany Paul and Elena. D. c To my family JZ
Acknowledgements To my family: those who put up with me in the past - Jo and Phil - and especially those who put up with me in the present - Margaret, Bethany, Paul, and Elena. D.C. To my family. J.Z. 7
Preface The objective of this book is to provide a simple introduction to mathemat ical analysis with applications in economics. There is increasing use of real and functional analysis in economics, but few books cover that material at an elementary level. Our rationale for writing this book is to bridge the gap between basic mathematical economics books(which deal with introductory calculus and linear algebra) and advanced economics books such as Stokey and Lucas Recursive Methods in Economic Dynamics that presume a work- ing knowledge of functional analysis. The major innovations in this book relative to classic mathematics books in this area(such as Royden's Real Analysis or Munkres' Topology) are that we provide:(i)extensive simple examples(we believe strongly that examples provide the intuition necessary to grasp difficult ideas); (ii) sketches of complicated proofs(followed by the complete proof at the end of the book); and (ii) only material that is rel evant to economists(which means we drop some material and add other topics(e.g. we focus extensively on set valued mappings instead of just point valued ones). It is important to emphasize that while we aim to make this material as accessible as possible, we have not excluded demanding mathe- matical concepts used by economists and that the book is self-contained ( virtually any theorem used in proving a given result is itself proven in our book) Road Map Chapter 1 is a brief introduction to logical reasoning and how to construct direct versus indirect proofs. Proving the truth of the compound statement If A, then B"captures the essence of mathematical reasoning; we take the truth of statement "A"as given and then establish logically the truth of statement"B"follows. We do so by introducing logical connectives and the
Preface The objective of this book is to provide a simple introduction to mathematical analysis with applications in economics. There is increasing use of real and functional analysis in economics, but few books cover that material at an elementary level. Our rationale for writing this book is to bridge the gap between basic mathematical economics books (which deal with introductory calculus and linear algebra) and advanced economics books such as Stokey and Lucasí Recursive Methods in Economic Dynamics that presume a working knowledge of functional analysis. The major innovations in this book relative to classic mathematics books in this area (such as Roydenís Real Analysis or Munkresí Topology) are that we provide: (i) extensive simple examples (we believe strongly that examples provide the intuition necessary to grasp difficult ideas); (ii) sketches of complicated proofs (followed by the complete proof at the end of the book); and (iii) only material that is relevant to economists (which means we drop some material and add other topics (e.g. we focus extensively on set valued mappings instead of just point valued ones)). It is important to emphasize that while we aim to make this material as accessible as possible, we have not excluded demanding mathematical concepts used by economists and that the book is self-contained (i.e. virtually any theorem used in proving a given result is itself proven in our book). Road Map Chapter 1 is a brief introduction to logical reasoning and how to construct direct versus indirect proofs. Proving the truth of the compound statement ìIf A, then Bî captures the essence of mathematical reasoning; we take the truth of statement ìAî as given and then establish logically the truth of statement îBî follows. We do so by introducing logical connectives and the 9
10 CONTENTS idea of a truth table We introduce set operations, relations, functions and correspondences n Chapter 2. Then we study the size"of sets and show the difference between countable and uncountable infinite sets. Finally, we introduce the notion of an algebra (just a collection of sets that satisfy certain properties) and"generate"(i.e. establish that there always exists) a smallest collection of subsets of a given set where all results of set operations (like complements union,and intersection) remain in the collection Chapter 3 focuses on the set of real numbers(denoted R), which is one of the simplest but most economic(both literally and figuratively) sets to introduce students to the ideas of algebraic, order, and completeness prop- erties. Here we expose students to the most elementary notions of distance open and closedness, boundedness, and simple facts like between any two real numbers is another real number One critical result we prove is the bolzano- Weierstrass Theorem which says that every bounded infinite subset of R has a point with sufficiently many points in any subset around it. This result has important implications for issues like convergence of a sequence of points which is introduced in more general metric spaces. We end by generating the smallest collection of all open sets in R known as the Borel(a-algebra In Chapter 4 we introduce sequences and the notions of convergence. com- leteness, compactness, and connectedness in general metric spaces, where we augment an arbitrary set with an abstract notion of a"distance"function Understanding these " C" properties are absolutely essential for economists For instance, the completeness of a metric space is a very important property for problem solving. In particular, one can construct a sequence of approxi- mate solutions that get closer and closer together and provided the space is complete, then the limit of this sequence erists and is the solution of the orig- inal problem. We also present properties of normed vector spaces and study two important examples, both of which are the used extensively in economics finite dimensional Euclidean space(denoted Rn) and the space of (infinite dimensional) sequences(denoted ep). Then we study continuity of functions and hemicontinuity of correspondences. Particular attention is paid to the properties of a continuous function on a connected domain(a generalization of the Intermediate Value Theorem)as well as a continuous function on a compact domain(a generalization of the Extreme Value Theorem). We end by providing fixed point theorems for functions and correspondences that are useful in proving, for instance, the existence of general equilibrium with competitive markets or a Nash Equilibrium of a noncooperative game
10 CONTENTS idea of a truth table. We introduce set operations, relations, functions and correspondences in Chapter 2 . Then we study the ìsizeî of sets and show the differences between countable and uncountable infinite sets. Finally, we introduce the notion of an algebra (just a collection of sets that satisfy certain properties) and ìgenerateî (i.e. establish that there always exists) a smallest collection of subsets of a given set where all results of set operations (like complements, union, and intersection) remain in the collection. Chapter 3 focuses on the set of real numbers (denoted R), which is one of the simplest but most economic (both literally and figuratively) sets to introduce students to the ideas of algebraic, order, and completeness properties. Here we expose students to the most elementary notions of distance, open and closedness, boundedness, and simple facts like between any two real numbers is another real number. One critical result we prove is the BolzanoWeierstrass Theorem which says that every bounded infinite subset of R has a point with sufficiently many points in any subset around it. This result has important implications for issues like convergence of a sequence of points which is introduced in more general metric spaces. We end by generating the smallest collection of all open sets in R known as the Borel (σ-)algebra. In Chapter 4 we introduce sequences and the notions of convergence, completeness, compactness, and connectedness in general metric spaces, where we augment an arbitrary set with an abstract notion of a ìdistanceî function. Understanding these ìCî properties are absolutely essential for economists. For instance, the completeness of a metric space is a very important property for problem solving. In particular, one can construct a sequence of approximate solutions that get closer and closer together and provided the space is complete, then the limit of this sequence exists and is the solution of the original problem. We also present properties of normed vector spaces and study two important examples, both of which are the used extensively in economics: finite dimensional Euclidean space (denoted Rn) and the space of (infinite dimensional) sequences (denoted !p). Then we study continuity of functions and hemicontinuity of correspondences. Particular attention is paid to the properties of a continuous function on a connected domain (a generalization of the Intermediate Value Theorem) as well as a continuous function on a compact domain (a generalization of the Extreme Value Theorem). We end by providing fixed point theorems for functions and correspondences that are useful in proving, for instance, the existence of general equilibrium with competitive markets or a Nash Equilibrium of a noncooperative game
CONTENTS Chapter 5 focuses primarily on Lebesgue measure and integration since almost all applications that economists study are covered by this case and because it is easy to conceptualize the notion of distance through that of the restriction of an outer measure. We show that the collection of lebesgue measurable sets is a o-algebra and that the collection of borel sets is a subset of the Lebesgue measurable sets. Then we provide a set of convergence theorems for the existence of a Lebesgue integral which are applicable under a wide variety of conditions. Next we introduce general and signed measures where we show that a signed measure can be represented simply by an integral the Radon-Nikodyn Theorem). To prepare for the following chapter, we end by studying a simple function space(the space of integrable functions) and prove it is complete. We study properties such as completeness and compactness in two impor- tant function spaces in Chapter 6: the space of bounded continuous functions (denoted C(X)) and the space of p-integrable functions(denoted Lp(X)).A fundamental result on approximating continuous functions in C(X) is given in a very general set of Theorems by Stone and Weierstrass. Also, the Brouwer Fixed Point Theorem of Chapter 4 on finite dimensional spaces is generalized to infinite dimensional spaces in the Schauder Fixed Point Theorem. Mov ing onto the Lp(X) space, we show that it is complete in the Riesz-Fischer Theorem. Then we introduce linear operators and functionals, as well as the notion of a dual space. We show that one can construct bounded linear functionals on a given set X in the Hahn-Banach Theorem, which is used to prove certain separation results such as the fact that two disjoint convex sets an be separated by a linear functional. Such results are used extensively in economics; for instance, it is employed to establish the Second Welfare Theorem. The chapter ends with nonlinear operators and focuses particu- larly on optimization in infinite dimensional spaces. First we introduce the weak topology on a normed vector space and develop a variational method of optimizing nonlinear functions. Then we consider another method of finding the optimum of a nonlinear functional by dynamic programming Chapter 7 provides a brief overview of general topological spaces and the idea of a homeomorphism (i.e. when two topological spaces X and Y have"similar topological structure"which occurs when there is a one-to-one and onto mapping f from elements in X to elements in Y such that both f and its inverse are continuous). We then compare and contrast topological and metric properties, as well as touch upon the metrizability problem (i finding conditions on a topological space X which guarantee that there exists
CONTENTS 11 Chapter 5 focuses primarily on Lebesgue measure and integration since almost all applications that economists study are covered by this case and because it is easy to conceptualize the notion of distance through that of the restriction of an outer measure. We show that the collection of Lebesgue measurable sets is a σ-algebra and that the collection of Borel sets is a subset of the Lebesgue measurable sets. Then we provide a set of convergence theorems for the existence of a Lebesgue integral which are applicable under a wide variety of conditions. Next we introduce general and signed measures, where we show that a signed measure can be represented simply by an integral (the Radon-Nikodyn Theorem). To prepare for the following chapter, we end by studying a simple function space (the space of integrable functions) and prove it is complete. We study properties such as completeness and compactness in two important function spaces in Chapter 6: the space of bounded continuous functions (denoted C(X)) and the space of p-integrable functions (denoted Lp(X)). A fundamental result on approximating continuous functions in C(X) is given in a very general set of Theorems by Stone and Weierstrass. Also, the Brouwer Fixed Point Theorem of Chapter 4 on finite dimensional spaces is generalized to infinite dimensional spaces in the Schauder Fixed Point Theorem. Moving onto the Lp(X) space, we show that it is complete in the Riesz-Fischer Theorem. Then we introduce linear operators and functionals, as well as the notion of a dual space. We show that one can construct bounded linear functionals on a given set X in the Hahn-Banach Theorem, which is used to prove certain separation results such as the fact that two disjoint convex sets can be separated by a linear functional. Such results are used extensively in economics; for instance, it is employed to establish the Second Welfare Theorem. The chapter ends with nonlinear operators and focuses particularly on optimization in infinite dimensional spaces. First we introduce the weak topology on a normed vector space and develop a variational method of optimizing nonlinear functions. Then we consider another method of finding the optimum of a nonlinear functional by dynamic programming. Chapter 7 provides a brief overview of general topological spaces and the idea of a homeomorphism (i.e. when two topological spaces X and Y have ìsimilar topological structureî which occurs when there is a one-to-one and onto mapping f from elements in X to elements in Y such that both f and its inverse are continuous). We then compare and contrast topological and metric properties, as well as touch upon the metrizability problem (i.e. finding conditions on a topological space X which guarantee that there exists
12 CONTENTS a metric on the set X that induces the topology of X Uses of the book We taught this manuscript in the first year PhD core sequence at the Uni- versity of Pittsburgh and as a Phd class at the University of Texas. The program at University of Pittsburgh begins with an intensive, one month re- medial summer math class that focuses on calculus and linear algebra. Our manuscript was used in the Fall semester class. Since we were able to quickly explain theorems using sketches of proofs, it was possible to teach the entire book in one semester. If the book was used for upper level undergradu ates, we would suggest simply to teach Chapters 1 to 4. While we used the manuscript in a classroom, we expect it will be beneficial to researchers; for instance, anyone who reads a book like Stokey and Lucas' Recursive Meth- ods must understand the background concepts in our manuscript. In fact, it was because one of the authors found that his students were ill prepared to understand Stokey and Lucas in his upper level macroeconomics class, that this project began
12 CONTENTS a metric on the set X that induces the topology of X). Uses of the book We taught this manuscript in the first year PhD core sequence at the University of Pittsburgh and as a PhD class at the University of Texas. The program at University of Pittsburgh begins with an intensive, one month remedial summer math class that focuses on calculus and linear algebra. Our manuscript was used in the Fall semester class. Since we were able to quickly explain theorems using sketches of proofs, it was possible to teach the entire book in one semester. If the book was used for upper level undergraduates, we would suggest simply to teach Chapters 1 to 4. While we used the manuscript in a classroom, we expect it will be beneficial to researchers; for instance, anyone who reads a book like Stokey and Lucasí Recursive Methods must understand the background concepts in our manuscript. In fact, it was because one of the authors found that his students were ill prepared to understand Stokey and Lucas in his upper level macroeconomics class, that this project began
Chapter 1 Introduction In this chapter we hope to introduce students to applying logical reasoning to prove the validity of economic conclusions(B)from well-defined premises (A). For example, A may be the statement " An allocation-price pair (, p) is a Walrasian equilibrium"and B the statement "the allocation is Pareto efficient". In general, statements such as A and or B may be true or false 1.1 Rules of logic In many cases, we will be interested in establishing the truth of statements of the form"If A, then B. Equivalently, such a statement can be written as:“A→B”;“ A implies”;“ A only if E”;“ a is sufficient for B”;or“Bis for A "Applied to the example given in the previous paragraph, "If A, then B"is just a statement of the First Fundamental Theorem of Welfare Economics. In other cases. we will be interested in the truth of statements of the form"A if and only if B. equivalently, such a statement can be written A”;“ A is necessary and sufficient for B";or“ A Is equivalent to2’ plies “A→ B and B→A” which is just“AB”;“ A implies盛 and B implies Notice that a statement of the form“A→B” is simply a construct of two simple statements connected by =>" Proving the truth of the statement A=B captures the essence of mathematical reasoning; we take the truth of A as given and then establish logically the truth of B follows. Before actually setting out on that path, let us define a few terms. A Theorem on Proposition is a statement that we prove to be true. A Lemma is a theorem we use to prove another theorem. A Corollary is a theorem whose proof is
Chapter 1 Introduction In this chapter we hope to introduce students to applying logical reasoning to prove the validity of economic conclusions (B) from well-defined premises (A). For example, A may be the statement ìAn allocation-price pair (x, p) is a Walrasian equilibriumî and B the statement ì the allocation x is Pareto efficientî. In general, statements such as A and/or B may be true or false. 1.1 Rules of logic In many cases, we will be interested in establishing the truth of statements of the form ìIf A,then B.î Equivalently, such a statement can be written as: ìA ⇒ Bî; ìA implies Bî; ìA only if Bî; ìA is sufficient for Bî; or ìB is necessary for A.î Applied to the example given in the previous paragraph, ìIf A,then Bî is just a statement of the First Fundamental Theorem of Welfare Economics. In other cases, we will be interested in the truth of statements of the form ìA if and only if B.î Equivalently, such a statement can be written: ìA ⇒ B and B ⇒ Aî which is just ìA ⇔ Bî; ìA implies B and B implies Aî; ìA is necessary and sufficient for Bî; or ìA is equivalent to B.î Notice that a statement of the form ìA ⇒ Bî is simply a construct of two simple statements connected by ì⇒î. Proving the truth of the statement ìA ⇒ Bî captures the essence of mathematical reasoning; we take the truth of A as given and then establish logically the truth of B follows. Before actually setting out on that path, let us define a few terms. A Theorem or Proposition is a statement that we prove to be true. A Lemma is a theorem we use to prove another theorem. A Corollary is a theorem whose proof is 13
