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: Logical sentences, negation, conjunction
Lecture 2: Disjunction, De Morgan's laws
Lecture 3: Distributive laws, implication
Lecture 4: Conditional and biconditional sentences, converse and contrapositive
Lecture 5: Conditional proof, tautologies, quantifiers
Lecture 6: Quantifiers
Lecture 7: Generalized De Morgan and distributive laws, order of quantifiers
Lecture 8: Unique existence, mathematical induction
Lecture 9: Induction
Lecture 10: Parity
Lecture 11: Axioms for the integers (also see the Integers handout)
Lecture 12: Basic consequences of the integer axioms, positive integers, inequalities
Lecture 13: The well-ordering principle, divisibility
Lecture 14: Divisibility
Lecture 15: Primes
Lecture 16: The infinitude of primes
Lecture 17: Division Algorithm
Lecture 18: Greatest common divisors, Euclidean Algorithm
Lecture 19: Reverse Euclidean Algorithm, congruence of integers
Lecture 20: Modular arithmetic
Lecture 21: More modular arithmetic
Lecture 22: The Fundamental Theorem of Arithmetic, the theorem on division by a prime
Lecture 23: Applications of FTA
Lecture 24: Proof of FTA
Lecture 25: Real, rational, and irrational numbers
Lecture 26: Square roots of integers
Lecture 27: Sets
Lecture 28: Subsets, algebra of sets
Lecture 29: DeMorgan's laws for sets, sets of sets, generalized unions and intersections
Lecture 30: Power sets, ordered pairs, Cartesian products
Lecture 31: Functions, graphs, function composition
Lecture 32: Surjections, injections, bijections (Zoom notes)
Lecture 33: Inverse functions
Lecture 34: More about inverse functions, cardinality
Lecture 35: Finite sets, pigeonhole principle
Lecture 36: Infinite sets, countability
Lecture 37: Uncountability of R