Topics

Lecture Download 
Introduction: course policies; Overview, Logic, Propositions

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Tautologies, Logical Equivalences

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Predicates and Quantifiers: "there exists" and "for all"

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Sets: curly brace notation, cardinality, containment, empty set {, power
set P(S), Ntuples and Cartesian product. Set Operations: set
operations union and disjoint union, intersection, difference,
complement, symmetric difference

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Functions: domain, codomain, range; image, preimage; onetoone, 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.

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BigOh, BigOmega, BigTheta: BigOh/Omega/Theta notation,
algorithms, pseudocode, complexity.

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Integers: Divisors Primality Fundamental Theorem of Arithmetic.
Modulii: Division Algorithm, Greatest common divisors/least common
multiples, Relative Primality, Modular arithmetic, Caesar Cipher,

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Number Theoretic Algorithms: Euclidean Algorithm for GCD; Number
Systems: Decimal, binary numbers, others bases;

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RSA Cryptography: General Method, Fast Exponentiation, Extended
Euler Algorithm, Modular Inverses, Exponential Inverses,
Fermat's Little Theorem, Chinese Remainder Theorem

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readme

Proof Techniques.

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Induction Proofs: Simple induction, strong induction, program
correctness

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Recursion: Recursive Definitions, Strings, Recursive Functions.

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Counting Fundamentals: Sum Rule, Product Rule,
InclusionExclusion, Pigeonhole Principle Permutations.

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rpermutations: P(n,r),
rcombinations: C(n,r),
Anagrams, Cards and Poker; Discrete probability: NY State Lotto,
Random Variables, Expectation, Variance, Standard Deviation.

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Stars and Bars.

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Recurrence Relations: linear recurrence relations with constant
coefficients, homogeneous and nonhomogeneous, nonrepeating and
repeating roots; Generelized IncludsionExclusion: counting onto
functions, counting derangements

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Representing Relations: Subsets of Cartesian products, Column/line
diagrams, Boolean matrix, Digraph; Operations on Relations: Boolean,
Inverse, Composition, Exponentiation, Projection, Join

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Graph theory basics and definitions: Vertices/nodes, edges,
adjacency, incidence; Degree, indegree, outdegree; Degree, indegree,
outdegree; 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.

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Connectedness, Euler and Hamilton Paths

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Planar Graphs, Coloring

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Reading Period. Review session TBA.

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