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
Lecture 6: Quantifiers
Lecture 7: Generalized De Morgan and distributive laws
Lecture 8: Order of quantifiers, unique existence
Lecture 9: Induction
Lecture 10: More induction, parity
Lecture 11: Axioms for the integers (also see the Integers handout)
Lecture 12: Some basic consequences of the integer axioms
Lecture 13: Positive integers and the well-ordering principle
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: The Fundamental Theorem of Arithmetic
Lecture 22: Theorem on division by a prime, applications of FTA
Lecture 23: Proof of FTA
Lecture 24: Real and rational numbers
Lecture 25: Irrational numbers
Lecture 26: Square roots of integers, intro to sets
Lecture 27: Sets, subsets
Lecture 28: Algebra of sets, DeMorgan's laws for sets
Lecture 29: Sets of sets, generalized unions and intersections, power sets, ordered pairs
Lecture 30: Cartesian products, functions
Lecture 31: Graphs, function composition
Lecture 32: Surjections, injections, bijections
Lecture 33: Inverse functions
Lecture 34: Cardinality
Lecture 35: Finite sets, pigeonhole principle
Lecture 36: Infinite sets, countability
Lecture 37: Uncountability of R