Chapter Zero

Introduction-an Essay. Mathematical Reasoning. Deciding What to Assume. What Is Needed to Do Mathematics? Chapter Zero. Logic. Statements and Predicates. Mathematical Implication. Direct Proofs. Compound Statements and Truth Tables. Equivalence. Proof by Contrapositive. Negating Statements. Proof by Contradiction. Existence and Uniqueness. Proving Theorems: What Now? Sets. Sets and Set Notation. Set Operations. Russells Paradox. Relations and Ordering. Relations. Orderings. Equivalence Relations. Functions. Basic Ideas. Composition and Inverses. Order Isomorphisms. Sequences. Binary Operations. Induction. Inductive Reasoning and Mathematical Induction. Using Induction. Complete Induction. Elementary Number Theory. Natural Numbers and Integers. Divisibility in the Integers. The Euclidean Algorithm. Relatively Prime Integers. Prime Factorization. Congruence Modulo n. Divisibility Modulo n. Cardinality. Galileos Paradox. Infinite Sets. Countable Sets. Beyond Countability. Comparing Cardinalities. The Continuum Hypothesis. Order Isomorphisms (Revisited). The Real Numbers. Constructing the Axioms. Arithmetic. Order. The Least Upper Bound Axiom. Sequence Convergence in R. Axiomatic Set Theory. Elementary Axioms. The Axiom of Infinity. Axioms of Choice and Substitution. Constructing R. From N to Integers. From Integers to Rationals. From Rationals to R. Index.