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The sequel to the course Discrete Mathematics: Open Doors to Great Careers.

Would you like to learn a mathematics subject that is crucial for many highdemand lucrative career fields such as:
If you're looking to gain a solid foundation in Discrete Mathematics, allowing you to study on your own schedule at a fraction of the cost it would take at a traditional university, to further your career goals, this online course is for you. If you're a working professional needing a refresher on discrete mathematics or a complete beginner who needs to learn Discrete Mathematics for the first time, this online course is for you.
Why you should take this online course: You need to refresh your knowledge of discrete mathematics for your career to earn a higher salary. You need to learn discrete mathematics because it is a required mathematical subject for your chosen career field such as computer science or electrical engineering. You intend to pursue a masters degree or PhD, and discrete mathematics is a required or recommended subject.
Why you should choose this instructor: I earned my PhD in Mathematics from the University of California, Riverside. I have extensive teaching experience: 6 years as a teaching assistant at University of California, Riverside, four years as a faculty member at Western Governors University, #1 in secondary education by the National Council on Teacher Quality, and as a faculty member at Trident University International.
In this course, I cover core topics such as:
After taking this course, you will feel CAREFREE AND CONFIDENT. I will break it all down into bitesized nobrainer chunks. I explain each definition and go through each example STEP BY STEP so that you understand each topic clearly. I will also be AVAILABLE TO ANSWER ANY QUESTIONS you might have on the lecture material or any other questions you are struggling with.
Practice problems are provided for you, and detailed solutions are also provided to check your understanding.
30 day full refund if not satisfied.
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Section 1: Introduction  

Lecture 1 
Introduction Lecture
Preview

03:11  
Section 2: Set Theory  
Lecture 2  09:19  
Students will be introduced to the notions of set, element, set equality, subset, and proper subset. 

Lecture 3  16:19  
Students will learn how to find the union, intersection, difference, and complements of sets. 

Lecture 4 
Problem Set: Operations on Sets

00:01  
Lecture 5  12:06  
Students will learn how to find the union of any number of sets and the intersection of any number of sets. 

Lecture 6 
Problem Set: General Unions and Intersections

00:01  
Lecture 7  07:24  
The notion of a partition is introduced. Students will be able to determine if a collection of sets forms a partition of a given set. 

Lecture 8 
Problem Set: Partitions

00:00  
Lecture 9  03:03  
The notion of the power set of a set is introduced. Students will be able to find the power set of a given set. 

Lecture 10  06:06  
The notion of the Cartesian product of a collection of sets is introduced. Students will be able to find the Cartesian product of a collection of sets. 

Lecture 11 
Problem Set: Power Set and Cartesian Product

00:01  
Lecture 12  10:36  
In this lecture, students are introduced to set identities. Students will be able to prove set identities. 

Lecture 13 
Problem Set: Set Identities

00:01  
Section 3: Functions  
Lecture 14  11:27  
The notion of a function is introduced. Students will understand what the domain and range of a function are, and students will understand what the image of an element is and what the preimage of an element is. 

Lecture 15  13:35  
In this lecture, we look at some examples of functions, including the floor function. Students will be able to find the domain and range of a function and the preimage of a given element in the codomain. 

Lecture 16 
Problem Set: Functions

00:00  
Lecture 17  11:30  
The onetoone and onto properties are introduced. Students will be able to prove that a function is onetoone or that a function is not onetoone. Students will also be able to prove that a function is onto or that a function is not onto. 

Lecture 18 
Problem Set: One to One and Onto Properties

00:02  
Lecture 19  17:32  
The notion of a onetoone correspondence and the notion of the inverse function of a given function are introduced. Students will be able to prove that a function is a onetoone correspondence. Students will also be able to find the inverse function of a given function. 

Lecture 20 
Problem Set: Onetoone Correspondence and Inverse Functions

00:01  
Lecture 21  14:39  
The notion of the composition of two functions is introduced. Students will be able to find the composition of two functions. 

Lecture 22 
Problem Set: Composition of Functions

00:01  
Lecture 23  09:02  
In this lecture, we apply functions to the task of measuring the sizes of sets. Students will be able to show that two sets have the same cardinality. 

Lecture 24 
Problem Set: Cardinality

00:00  
Lecture 25  15:22  
The notion of a countable set is introduced. An example of an uncountable set is discussed. Students will be able to show that a set is countable or that a set is uncountable. 

Lecture 26  09:34  
In this lecture, an additional example of an uncountable set is discussed. The Cantor diagonalization process is also introduced. Students will be able to apply the Cantor diagonalization process to show that a set is uncountable. 

Lecture 27 
Problem Set: Countable Sets, Uncountable Sets, Cantor Diagonalization Process

00:02  
Section 4: Relations  
Lecture 28  05:43  
The notion of a relation is introduced. Students will understand what the domain and codomain of a relation are, and students will understand what it means for x to be related to y. 

Lecture 29  09:15  
We explore additional examples of relations. The notion of a relation on a set A is introduced. Also, the notion of congruence modulo 5 is introduced. 

Lecture 30 
Problem Set: Relations

00:00  
Lecture 31  05:52  
In this lecture, we explore relationships between 3 or more things. Students will see examples of nary relations. 

Lecture 32  08:38  
The properties of reflexivity, symmetry, and transitivity are introduced. Students will be able to determine if a given relation satisfies these properties. 

Lecture 33  09:18  
We explore the properties of the relation of congruence modulo 5. Students will further reinforce their understanding of the properties of relations. 

Lecture 34 
Problem Set: Properties of Relations

00:01  
Lecture 35  07:14  
The notion of equivalence relation is introduced. Students will be able to prove that a given relation is an equivalence relation. 

Lecture 36 
Problem Set: Equivalence Relations

00:01  
Lecture 37  04:21  
The notion of a partition induced by an equivalence relation is introduced. The notion of an equivalence class is also introduced. Students will be able to find the partition induced by an equivalence relation on a set. 

Lecture 38  14:13  
We explore the partition induced by congruence modulo 5 on the set of integers. 

Lecture 39 
Problem Set: Partition Induced by an Equivalence Relation

00:02  
Lecture 40  06:28  
The notion of an equivalence relation induced by a partition is introduced. 

Lecture 41 
Problem Set: Equivalence Relation Induced by a Partition

00:02  
Section 5: Partial Order Relations  
Lecture 42  09:12  
The notion of a partial order relation is introduced. Students will understand the properties of a partial order relation and what it means for two elements to be comparable. 

Lecture 43 
Problem Set: Partial Order Relations

00:01  
Lecture 44  03:54  
We explore the 'prerequisite' relation on a set of math courses. Students will be able to represent a partially ordered set using a directed graph. 

Lecture 45  06:18  
The notion of a Hasse diagram of a partial order relation is introduced. The notions of a maximal and of a minimal element are also introduced. Students will be able to represent a partially ordered set using a Hasse diagram. 

Lecture 46 
Problem Set: Hasse Diagrams

00:01  
Lecture 47  03:38  
The notion of a topological sorting for a partial order relation is introduced. 

Lecture 48  10:08  
In this lecture, we find a topological sorting for the prerequisite relation on a set of math courses. Students will be able to find a topological sorting for a given partial order relation. 

Lecture 49 
Problem Set: Topological Sorting

00:01  
Section 6: Algorithms  
Lecture 50  08:32  
In this lecture, we explore a simple example of an algorithm. 

Lecture 51  07:41  
The definition of an algorithm is introduced, and the while loop is introduced. The general form of an algorithm is also explained. Students will understand while loops and the general form of an algorithm. 

Lecture 52 
Problem Set: Algorithms

00:00  
Lecture 53  03:16  
The definition of correctness of a while loop is introduced. Students will understand what it means for a while loop to be correct with respect to its precondition and postcondition. 

Lecture 54  06:34  
The notion of a loop invariant is introduced, and the Loop Invariant Theorem is introduced. Students will understand how a loop invariant is used to prove the correctness of a while loop. 

Lecture 55  16:40  
We prove the correctness of the division algorithm. Students will be able to prove the correctness of a given algorithm. 

Lecture 56 
Problem Set: Correctness of Algorithms

00:01  
Lecture 57  01:22  
The definition of the efficiency of an algorithm is introduced. Students will understand what the efficiency of an algorithm means. 

Lecture 58  11:30  
We explore the linear search algorithm. 

Lecture 59  07:56  
In this lecture, we explore the efficiency of the linear search algorithm. The notions of bestcase efficiency and worstcase efficiency are introduced. Students will be able to measure the efficiency of algorithms. 

Lecture 60  03:04  
The notion of the order of a function is introduced. Students will be able to determine the order of a given function. 

Lecture 61  05:13  
We explore the binary search algorithm. 

Lecture 62  03:30  
In this lecture, we explore the efficiency of the binary search algorithm. Students will reinforce their ability to measure the efficiency of algorithms. Students will understand what it means for an algorithm to be more efficient than another algorithm. 

Lecture 63 
Problem Set: Efficiency of Algorithms

00:01  
Section 7: Concluding Letter  
Lecture 64 
Concluding Letter

00:20  
Lecture 65 
Bonus Lecture

00:09 
Hi there! My name is Richard Han. I earned my PhD in Mathematics from the University of California, Riverside. I have extensive teaching experience: 6 years as a teaching assistant at University of California, Riverside, over two years as a faculty member at Western Governors University, #1 in secondary education by the National Council on Teacher Quality, and as a faculty member at Trident University International. My expertise includes calculus and linear algebra. I am an instructor on Udemy for the courses Philosophy of Language: Solidify Critical Thinking Skills and Linear Algebra for Beginners: Open Doors to Great Careers.