# Universite of Cincinnati Discrete Math

On Bilibili

## Basic Notation Introdution

• 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
• 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)