[ Learn ] Cincinnati 大学离散数学课程笔记
Universite of Cincinnati Discrete Math
对目录的理解是最基本的，把课程的目录整理出来，对整个课程需要讲解的东西有一个整体的认识。进而在细节上琢磨.
印象会更加深刻.
Basic Notation Introdution
这一节当中介绍了最基本的研究单位 Set,集合
 Introduction of Discrete Math
 Intro to Set
 SetRoster
{}
vs SetBuilder{xP(x)}
notation  The Empty Set & Vacuous Truth
 Cartesion Product of Two Sets AxB
 Relation Between two sets
 The intuitive idea of function
 Formal Definition of Fuction using the Cartesion Product
 Example: is this relation a function? x^2 + y^2 = 1
Logical Statements
 Intro to Logical Stagements

Intro to Truth Tables(Nagation,Conjuction,Disjuction: ~,&, ) 
Truth Table ~p ~q  Logical Equivalence of Two Statements
 Tautologies(赘述) and Contradictions(矛盾)
 3 ways to show a Logical Equivalence(DeMorgan’s Law)
 Conditional Statements( if P then q: p > q)
 Vacuously True Statements
 Nagating a Conditional Statement
 Contrapositive(对换句) of a Conditional Statement
 The converse and inverse of a Conditional statement
 Biconditional Statements(if and only if: p <> q)
 Logical Arguments(Modus Ponens & Modus Tollens)
 Logical Argument Forms( Generalizations, Specialization, contradiction)
 Analyzing and argument for validity
 Predicates and their Truth Sets
 Universal and Existential Quantifiers( For All: A / There Exists: E)
 Nagating Universal and Existential Quantifiers
 Nagating Logical Statements with Multiple Quantifiers
 Universal Conditionals P(x) implies Q(x)
 Necessary and Sufficient Conditions
 Formal Definitions in Math(Ex: Even & Odd Integers)
Proof
 How to Prove Math Theorems(1st Ex: Even + Odd = Odd )
 StepByStep Guid to Proofs(Ex: sum of two evens is even)
 Retional Numbers(Definition + FIrst Proof)
 Proving that divisibility is transitive
 Disprovisng implications with Counter examples
 Proof by Division Into Cases
 Proof by Contradiction( Mathod & First Example)
 Proof By Contraposive( Mathod & First Example)
 QuotientRemainder Theorem and Modular Arithmetic
 Proof(There are infinitely many primes numbers)
Sequences
 Introduction to Sequences
 The Formal definition of a sequence
 The sum and product of finite sequences
 Introduction to Mathematical Induction
 Induction Proofs Involving Inequalities
 Strong Induction
 Recursive Sequences
 The Miraculous Fibonacci Sequence
 Prove A is a subset of B with
element method
 Proving enqualities of sets using the element method
 The union of two sets
 The Intersection of Two Sets
 Universes and Compliments in Set Theory
 Using the Element Method to Prove a Set Containment w Modus Tollens
 Relations and their inverses
 Reflexive, Symmetric, and Transitive on a Set
 You need to check every spot for reflexivity, symmetry, and transitivity
 Equivalence Relations
Probablity
 Introduction to probablity
 Example: Computing the Probablity on independent events
 What is the probability of gussing a 4 digit pin code?
 Permutations ( How Many ways to rearrange the letters in a word?)
 The summation rule for disjoint unions
 Counting when the sample space is a nondisjoint union
 Counting the number of ways to choose r items from n items
 How many ways are there to reorder the word MISSISSIPPI?
 Counting and Probability Walthrough
 Introduction to Conditional Probability
 Two Conditional Probability Examples
 Conditional Probability With Tables( change of an Orange M&M)
 Bayes’ Theorem(The Simplest Case)
 Bayes’ Theorem Example(Surprising False Positives)