A Transition to Advanced Mathematics

1. LOGIC AND PROOFS. Propositions and Connectives. Conditionals and Biconditionals. Quantifiers. Basic Proof Methods I. Basic Proof Methods II. Proofs Involving Quantifiers. Additional Examples of Proofs 2. SET THEORY. Basic Notions of Set Theory. Set Operations. Extended Set Operations and Indexed Families of Sets. Induction. Equivalent Forms of Induction. Principles of Counting. 3. RELATIONS AND PARTITIONS. Relations. Equivalence Relations. Partitions. Ordering Relations. Graphs. 4. FUNCTIONS. Functions as Relations. Constructions of Functions. Functions That Are Onto; One-to-One Functions. One-to-One Correspondences and Inverse Functions. Images of Sets. Sequences. 5. CARDINALITY. Equivalent Sets; Finite Sets. Infinite Sets. Countable Sets. The Ordering of Cardinal Numbers. Comparability of Cardinal Numbers and the Axiom of Choice. 6. CONCEPTS OF ALGEBRA: GROUPS. Algebraic Structures. Groups. Subgroups. Operation Preserving Maps. Rings and Fields. 7. CONCEPTS OF ANALYSIS: COMPLETENESS OF THE REAL NUMBERS. Ordered Field Properties of the Real Numbers. The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem. The Bounded Monotone Sequence Theorem. Comparability of Cardinals and the Axiom of Choice.