Universite of Cincinnati Discrete Math

On Bilibili

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
  • Set-Roster {} vs Set-Builder {x|P(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 )
  • Step-By-Step 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)
  • Quotient-Remainder 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)