Text: Discrete Mathematics and its Applications, 6th edition, by Kenneth H. Rosen.
1 The Foundations: Logic and Proofs
Topics in chapter 1: proposition, truth value, negation, logical operators, compound proposition, truth table, disjuction, conjunction, exclusive or, implication, converse, contrapositive, bit, Boolean variable, bit operation, bit string, bitwise operation; tautology, contradiction, contingency, logical equivalnce, propsitional function De Morgan's laws; predicates, existential quantifier, universal quantifier; nested quantifiers, free & bound variables; rules of inference; theorem, conjecture, proof, lemma, corollary, fallacy, circular reasoning (begging the question), vacuous & trivial proof, direct & indirect proof; proof by cases, counterexample.
2 Basic Structures: Sets, Functions, Sequences, and Summations
Topics in chapter 2: set, axiom, paradox, element, empty set, set equality, subset, finite & infinite set, cardinality, power set,products of sets; union, intersection, disjoint sets, set difference, set complement, symmetric difference, Venn diagrams; function, domain, codomain, image, preimage, range, onto function, 1-1 function, 1-1 correspondence, inverse function, composition, floor, ceiling; sequence, string, summation notation, product notation, countable & uncountable set.
3 The Fundamentals: Algorithms, the Integers, and Matrices
Topics in chapter 3: algorithm, searching algorithm, linear search algorithm, binary search algorithm, time complexity, space complexity, worst-case time complexity, average-case time complexity; divisibility, prime & composite number, Mersenne prime, greatest common divisor, relatively prime, pairwise relatively prime integers, least common multiple, remainder & modulus, encryption & decryption, binary representation, hexadecimal represention, linear combination, inverse modulo n, linear congruence, pseudoprime, private and public key encryption; Euclidean algorithm
4 Induction, and Recursion
Topics in chapter 4: mathematical induction; recursive defined functions, sets, and structures; recursive algorithms; iteration.
5 Counting
6 Discrete Probability
8 Relations