These notes are the basis for our Math 3345 lectures. There may be material discussed in class which is not included in these notes, and vice versa.
Lecture 1: Setting the stage, logical sentences
Lecture 2: Negation, conjunction, disjunction, De Morgan's laws
Lecture 3: Distributive laws, implication
Lecture 4: Conditional and biconditional sentences, converse and contrapositive
Lecture 5: Conditional proof
Lecture 6: Tautologies, quantifiers
Lecture 7: Quantifiers, generalized De Morgan's laws
Lecture 8: Generalized distributive laws, order of quantifiers
Lecture 9: Order of quantifiers, unique existence
Lecture 10: Induction
Lecture 11: Parity
Lecture 12: Axioms for the integers
Lecture 13: The well-ordering principle
Lecture 14: Divisibility
Lecture 15: Primes
Lecture 16: The infinitude of primes
Lecture 17: Real and rational numbers
Lecture 18: Irrational numbers
Lecture 19: Division Algorithm
Lecture 20: Greatest common divisors, Euclidean Algorithm
Lecture 21: Reverse Euclidean Algorithm, congruence of integers
Lecture 22: Modular arithmetic
Lecture 23: The Fundamental Theorem of Arithmetic and applications
Lecture 24: Prime factorization and GCD computations, proof of FTA part 1
Lecture 25: Proof of FTA part 2, sets
Lecture 26: Set notation, subsets
Lecture 27: Union, intersection, relative complement
Lecture 28: DeMorgan's laws for sets, sets of sets, generalized unions and intersections
Lecture 29: More generalized unions and intersections, power sets, Cartesian products
Lecture 30: Functions
Lecture 31: Graphs, function composition
Lecture 32: Surjections, injections, bijections, inverse functions
Lecture 33: Inverse functions, cardinality
Lecture 34: Finite and infinite sets
Lecture 35: Countability and uncountability