MATH 112 • Tüm Sınavlar • Discrete Mathematics

Introduction

Negation of a Proposition

Compound Propositions

Truth Table 1

Truth Table 2

Tautology

Contradiction

Converse Inverse Contrapositive

Practice Problem 1

Practice Problem 2

Practice Problem 3

Logical Equivalences 1

Logical Equivalences 2

De Morgans' Law

Example 1

Example 2

Example 3

Example 4

Universal Quantifiers

Existential Quantifiers

Truth Value of Propositions with Quantifiers

Example 1

Example 2

Nested Quantifiers

Example 3

Example 4

Negation of Nested Quantifiers

Example 5

Translations

Example 6

Example 7

Exam-like Question 1

Translation of Statements With Multiple Variables

Valid Argument

Rules of Inferences

Example 1

Example 2

Example 3

Using Rules of Inferences to Build Arguments

Example 4

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Truth Table & Tautology (Fall 2022)

Logical Equivalances (Spring 2023)

Logical Equivalances (Spring 23: Make-Up)

Logical Equivalances (Summer 2020)

Logical Equivalances (Spring 2017)

Tautology (Fall 2022)

Quantifiers 1

Quantifiers 2

Quantifiers (Spring 2023)

Quantifiers (Spring 2023: Make-UP)

Quantifiers 3

Nested Quantifiers 1

Nested Quantifiers 2

Translations (Fall 2020)

Translations (Summer 2020)

Translations (Spring 2019)

Direct Proof

Example 1

Example 2

Example 3

Example 4

Proof by Contrapositive

Example 1

Example 2

Example 3

Proof by Contradiction

Example 1

Example 2

Proof by Counterexample

Example

Proofs of Equivalence

Example

Introduction

Proof of Formulas by Induction

Example 1

Example 2

Example 3

Example 4

Proof of Divisibility by Induction

Example 1

Example 2

Proof of Inequality by Induction

Example 1

Example 2

Induction Related to Fibonacci

Strong Induction

Direct Proof 1

Direct Proof 2

Direct Proof 3

Direct Proof 4

Proof by Contrapositive (Fall 2020)

Proof by Contrapositive

Proof by Contradiction 1

Proof by Contradiction 2

Proof by Contradiction (Fall 2022)

Proof by Contradiction (Fall 2020)

Proofs of Equivalence

Proof by Counterexample

Induction for Formulas 1

Induction for Formulas 2 (Fall 2017)

Induction for Formulas 3 (Summer 2020)

Induction for Formulas 4 (Spring 2023)

Induction for Formulas 5 (Spring 23: Make-Up)

Induction Proof for Divisibility 1

Induction Proof for Divisibility 2

Direct Proof & Induction (Spring 2023)

Direct Proof & Induction (Spring 23: Make-Up)

Induction for Inequalities 1

Induction for Inequalities 2 (Fall 2022)

Induction Proof for Inequalities 3

Induction with Fibonacci (Spring 2019)

Induction with Fibonacci (Fall 2020)

Definition and Notation

Subset

Example 1

Union and Intersection of Two Sets

Difference of Two Sets

Set Identities

Example 2

Power Set

Example 3

Cartesian Product

Example 4

Example 5

Proof Example 1

Proof Example 2

Definition of Function

Example 1

Number of Functions

Injective Functions

Example 2

Example 3

Example 4

Number of Injective Functions

Surjective Functions

Example 5

Example 6

Example 7

Bijective Function

Inverse Function

Example 8

Example 9

Proofs & Sets 1

Proofs & Sets 2 (Spring 2023)

Proofs & Sets 3 (Spring 2023 - Make Up)

Functions 1

Functions 2 (Spring 2023)

Functions 3 (Spring 2023 - Make Up)

Basic Principles of Counting

Counting Examples

Permutations

Permutations Example

Groups and Circular Permutation Example

Identical Objects Example 1

Identical Objects Example 2

Identical Objects Example 3

Combination

n choose r

Committee Example 1

Committee Example 2

Ball Example 1

Ball Example 2

Pigeonhole Principle

Example 1

Example 2

Example 3

Example 4

Example 5

Principlie Inclusion Exclusion

Example 1

Example 2

Binomial Theorem

Example 1

Example 2

Example 3

Pascal Identity

Combinatorial Proof

Example 4

Counting Standard Models 1

Counting Standard Models 2

Counting Standard Models 3

Counting Standard Models 4

Counting Standard Models 5

Counting Standard Models 6 (Spring 2023)

Counting Standard Models 7 (Spring 2023 - Make Up)

Counting Standard Models & Binomial Coefficients 1

Binomial Identities

Pigeonhole Principle 1

Pigeonhole Principle 2 (Spring 2023)

Pigeonhole Principle 3

Pigeonhole Principle 4

Pigeonhole Principle 5

Pigeonhole Principle 6

Introduction

Homogenous Recurrence Relations

Example 1

Example 2

Example 3

Example 4

Non Homogenous Recurrence Relations

Example5

Example 6

Creating Recurrence Relations

Example 7

Example 8

Example 9

Recurrence Relations 1 (Fall 2022)

Recurrence Relations 2 (Fall 2022)

Recurrence Relations 3 (Fall 2022)

Recurrence Relations 4

Recurrence Relations 5 (Spring 2023)

Recurrence Relations 6 (Fall 2022)

Recurrence Relations 7

Recurrence Relations 8 (Spring 2023)

Recurrence Relations 9

Definition

Example 1

Example 2

Example 3

Example 4

Extended Binomial Theorem

Counting Problems with Generating Functions

Example 5

Example 6

Example 7

Generating Functions 1 (Spring 2023)

Generating Functions 2 (Spring 2023)

Generating Functions 3 (Spring 2023)

Generating Functions 4

Generating Functions 5

Generating Functions 6

Generating Functions 7

Generating Functions 8

Generating Functions 9

Generating Functions 10

Generating Functions 11

Recurrence Relations

Recurrence Relations

Recurrence Relations

Recurrence Relations

Recurrence Relations

Generating Functions

Generating Functions

Generating Functions

Generating Functions

Combinatorial Proof

Generating Functions

Question 1

Question 2

Sample Space and Events

Probability

Axioms of Probability

Some Rules

Coin Example

Dice Example

Card Example

Ball Example

Set Example

Birthday Example

Conditioning Events

Total Probability Rule

Example 1

Example 2

Example 3

Example 4

Bayes' Rule

Bayes' Rule Example 1

Bayes' Rule Example 2

Independence

Independence Example 1

Independence Example 2

Random Variables

Probability Mass Function

PMF Example 1

PMF Example 2

Expected Value

Expected Value Example 1

Expected Value Example 2

Variance

Variance Example 1

Variance Example 2

Bernoulli Distribution Part 1

Bernoulli Distribution Part 2

Example 1

Example 2

Binomial Distribution Part 1

Binomial Distribution Part 2

Example 3

Example 4

Geometric Distribution Part 1

Geometric Distribution Part 2

Example 5

Example 6

Negative Binomial Distribution Part 1

Negative Binomial Distribution Part 2

Example 7

Example 8

Discrete Probability 1

Discrete Probability 2 (Spring 2023)

Discrete Probability 3 (Spring 2023)

Discrete Probability 4 (Spring 2023)

Discrete Probability 5 (Fall 2022)

Discrete Probability 6 (Spring 2023)

Conditional Probability and Independence 1

Conditional Probability and Independence 2

Conditional Probability and Independence 3

Conditional Probability and Independence 4

Bayes' Rule 1

Bayes' Rule 2

Bayes' Rule 3

Bayes' Rule 4

Discrete Random Variables 1

Discrete Random Variables 2

Binomial Distribution 1

Binomial Distribution 2

Binomial Distribution 3

Binomial Distribution 4

Negative Binomial Distribution 1

Negative Binomial - Geometric Distribution

Geometric Distribution 1

Geometric Distribution 2

Introduction

Graph Terminology

Handshaking Theorem

Example 1

Special Graphs

Example 2

Bipartite Graphs

Example 3

Example 4

Complete Bipartite Graph

Matching

Subgraph

Example 5

Subgraph Induced

Edge Contraction

Example 6

Complementary Graph

Adjacency Matrices - Undirected Graphs

Adjacency Matrices - Directed Graphs

Example 1

Example 2

Incidence Matrices

Example 3

Isomorphism of Graphs

Example 4

Example 5

Example 6

Example 7

Definition of Paths and Circuits

Connected Graphs

Euler Paths and Circuits

Example 8

Hamilton Paths and Circuits

Example 9

Example 10

Planar Graph

Planar Graph - Euler Formula

Example 11

Corollaries about Planar Graph

Example 12

Introduction

Rooted Tree

Terminology for Rooted Trees

Full m-ary Trees

Some Formulas

Adjacency Matrices

Incidence Matrices

Bipartite Graphs & Euler Circuits (Spring 2023)

Bipartite Graphs, Euler Circuit & Planar Graphs (Fall 2022)

Euler Path, Induced Subgraph & Incidence Matrix (Fall 2022)

Chromatic Number 1

Chromatic Number 2 (Spring 2023)

Isomorphism 1 (Spring 2023)

Isomorphism 2

Isomorphism 3 (Fall 2022)

Isomorphism 4 (Fall 2022)

Isomorphism 5 (Fall 2022)

True/False Problems 1 (Spring 2023)

True/False Problems 2 (Spring 2023)

Induced Subgraphs

Proof for Graphs

Proof for Graphs

Complete Bipartite Graphs

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18