Discrete Math Lecture Notes

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Introduction: course policies; Overview, Logic, Propositions
Tautologies, Logical Equivalences
Predicates and Quantifiers: "there exists" and "for all"
Sets: curly brace notation, cardinality, containment, empty set {, power set P(S), N-tuples and Cartesian product. Set Operations: set operations union and disjoint union, intersection, difference, complement, symmetric difference
Functions: domain, co-domain, range; image, pre-image; one-to-one, onto, bijective, inverse; functional composition and exponentiation; ceiling and floor. Sequences, Series, Countability: Arithmetic and geometric sequences and sums, countable and uncountable sets, Cantor's diagonilation argument.
Big-Oh, Big-Omega, Big-Theta: Big-Oh/Omega/Theta notation, algorithms, pseudo-code, complexity.
Integers: Divisors Primality Fundamental Theorem of Arithmetic. Modulii: Division Algorithm, Greatest common divisors/least common multiples, Relative Primality, Modular arithmetic, Caesar Cipher,
Number Theoretic Algorithms: Euclidean Algorithm for GCD; Number Systems: Decimal, binary numbers, others bases;
RSA Cryptography: General Method, Fast Exponentiation, Extended Euler Algorithm, Modular Inverses, Exponential Inverses, Fermat's Little Theorem, Chinese Remainder Theorem
Proof Techniques.
Induction Proofs: Simple induction, strong induction, program correctness
Recursion: Recursive Definitions, Strings, Recursive Functions.
Counting Fundamentals: Sum Rule, Product Rule, Inclusion-Exclusion, Pigeonhole Principle Permutations.
r-permutations: P(n,r), r-combinations: C(n,r), Anagrams, Cards and Poker; Discrete probability: NY State Lotto, Random Variables, Expectation, Variance, Standard Deviation.
Stars and Bars.
Recurrence Relations: linear recurrence relations with constant coefficients, homogeneous and non-homogeneous, non-repeating and repeating roots; Generelized Includsion-Exclusion: counting onto functions, counting derangements
Representing Relations: Subsets of Cartesian products, Column/line diagrams, Boolean matrix, Digraph; Operations on Relations: Boolean, Inverse, Composition, Exponentiation, Projection, Join
Graph theory basics and definitions: Vertices/nodes, edges, adjacency, incidence; Degree, in-degree, out-degree; Degree, in-degree, out-degree; Subgraphs, unions, isomorphism; Adjacency matrices. Types of Graphs: Trees; Undirected graphs; Simple graphs, Multigraphs, Pseudographs; Digraphs, Directed multigraph; Bipartite; Complete graphs, cycles, wheels, cubes, complete bipartite.
Connectedness, Euler and Hamilton Paths
Planar Graphs, Coloring
Reading Period. Review session TBA.

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