Example texts:
Mathematical Thinking: Problem-Solving and Proofs, D’Angelo, J. and D. West, 1999.
An Introduction to Abstract Mathematics, Donaldson, N. and A. Pantano. 2020.
List of topics:
- Introduction (1 day)
- Logic and Language of proofs (5 lectures)
- Logical statements and connectives; methods of proof; quantifiers; negation of qualifiers
- Divisibility and congruence (3 lectures)
- Euclidean algorithm, congruence classes
- Sets and functions (6 lectures)
- Set notation; proofs involving sets; functions; function composition; injectivity, surjectivity, bijectivity; inverse functions
- Induction (4 lectures)
- Induction; strong induction; well-ordering principle; many applications and examples
- Set theory (2 lectures)
- Cartesian products; power sets
- Relations and equivalence relations (4 days)
- Relations; equivalence relations; partitions; well-defined functions
- Cardinality (4 lectures)
- Countable sets, including integers and rationals; uncountable sets, including cardinality of the reals; Cantor-Schroder-Bernstein theorem (proof highly optional); cardinality of power sets; and many applications
- Analysis overview (8 lectures)
- Completeness; supremum and infimum; limits of sequences; monotone convergence theorem; limit laws; Cauchy sequences and subsequences; Bolzano-Weierstrass theorem and Cauchy’s criterion.
- Exams (3 lectures)
- Review, leeway (3 lectures)