Here is a basic outline of the topics we will cover (time permitting!) this semester.
- Fundamental Concepts
- Sets: Describing sets, products, unions, intersections and complements, Venn diagrams.
- Logic: Statements, compounds, conditional statements, quantifiers, negations.
- Counting: (optional) Lists, factorials, combinatorics, the inclusion-exclusion principle.
- Proving conditional statements
- Direct proof: Theorems and definitions, cases.
- Contrapositives: Proofs and exposition.
- Proof by contradiction: Proof and combining techniques in a proof.
- More Proofs
- Non-conditional statements: Equivalences, existence and uniqueness.
- Proofs and sets: Set inclusion, subsets, equality.
- Disproof: Counterexamples and contradiction.
- Induction: Mathematical induction, examples.
- Relations, functions and cardinality
- Relations: Relations, equivalence relations, partitions.
- Functions: Injectivity, surjectivity, compositions, inverses, images.
- Cardinality: Equal cardinality, countability, Cantor-Schroder-Bernstein Theorem.