**DISCRETE MATHEMATICS**

# Lecture Notes

by Zeph Grunschlag

**Click on the blue colored links to download the lectures.**

Topics | Lecture Download |

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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