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 game10 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 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 (σ-)algebra. In Chapter 4 we introduce sequences and the notions of convergence, com￾pleteness, 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 approxi￾mate 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 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 !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