A Taste of Pure Mathematics
by
Jerry Farlow
Chapter 1. Logic and Proofs
Section 1.2 Conditional and Biconditional Connectives
Section 1.4 Mathematical Proofs
Section 1.5 Proofs in Predicate Logic
Section 1.6 Proof by Mathematical Induction
Chapter 2. Sets
Section 2.1 Basic Operations of Sets
Section 2.4 Countable Infinity
Section 2.6 Cantor's Theorem and the ZFC Axioms
Chapter 3 Relations
Section 3.3 Equivalence Relations
Section 3.4 Basic Results of Graph Theory
Section 3.5 Directed Graphs: Tournament Graphs
Chapter 4: Functions
Section 4.1 The Function Relation
Section 4.2 Injections, Surjections, and Bijections
Section 4.3 Image and Inverse Image of a Set
Chapter 5: Analysis
Section 5.1: Construction of the Real Numbers
Section 5.2: The Complete Ordered Field: The Real Numbers
Section 5.3 General Topology on the Real Line: Open Sets
Section 5.4 The Bolzano-Weierstrass and Heine-Borel Theorems
Section 5.5 Uniformity and Compactness
Section 5.6 A Hint of Topology
Chapter 6: Algebra
Section 6.1 Symmetries and Algebraic Systems
Section 6.2 Introduction to the Algebraic Group
Section 6.3 Permutations Groups
Section 6.4 Subgroups: Groups within a Group