# Discrete Structures

**Course Code:**None

**Credits:**Noncredit

**Certificate Via Westcott**

Enroll Now (Noncredit)

View the Credit Option

**Credit Information**

Course Title: | Discrete Structures |

Course Code: | None |

Credits: | Noncredit |

Certificate Provider: | Westcott Courses |

Transfer: | No, the noncredit version of the course is not transferable. |

Grading Mode: | Standard letter grade |

**Course Requirements**

Enrollment Type: | Rolling | Sign up and start any day of the year. |

Requires Textbook? | No | All of the course materials are online. |

Proctored Final: | No | The noncredit version doesn't have a proctored final; but your college may require it. |

**Course Description**

This course was designed for students in math and computer science. Logic is emphasized in this course, and topics include proof and theory (including inductive and deductive proofs), propositional and predicate logic, set theory, algorithms (including recursion), trees, relations and functions, counting and probability, and elements of the theory of directed and undirected graphs (including Dijkstra's shortest path algorithm). Additionally, an introduction to complexity of algorithms and recurrence relations are included. Upon completion, students will be able to solve real world problems and use appropriate models for analysis.

Discrete Structures is an Omega Math

^{TM}Course.

**Prerequisite:**College Algebra or Precalculus with a grade of C or better.

**Course Pricing**

The full cost of the course is $509.00; however, you may use the payment plan to split the cost into two payments.

**Payment Plan:**

1st payment: $283.45 (Start your class immediately!)

2nd payment: $232.55 (Automatically charged 30 days later.)

Enroll Now - Noncredit - $509 View Credit Option - $728

**Helpful Links:**

Math Placement Test

View a free Omega Math Demo Lesson

See the Certificate & Preapproval tab for more details on this topic.

The noncredit course is the exact same class as the credit course; it is just less expensive since it is not sent through our partner university for credit. The noncredit courses can be used for remediation, personal enrichment, or students can learn the material and then receive credit at a home college using their college's Credit by Examination option.

**Enrollment Information**

Enroll any day of the year, and start that same day. Students have five months of access, plus a 30 day extension at the end if needed. Students can finish the self-paced courses as soon as they are able, with a minimum coursework time of at least four weeks. Most students finish the lower level courses in four weeks to three months. The upper level math courses, such as calculus and above, usually take students 2 - 4 months.

**Textbook Requirement:**

No outside textbook is needed. Our Omega Math

^{TM}courses contain all the lessons, homework, solution manuals, quizzes, tests and the final. Our lessons start out with the easiest examples, and then move slowly to the more advanced problems. Between examples, there are interactive problems which make sure the student understands the concepts, as well as enables the student to store the information into long term memory.

**Proctored Final Requirement:**

A proctored final is not required for the noncredit courses. However, many colleges permit their students to use our noncredit certificate as a prerequisite for the next level course. These colleges require a proctored final for the course. If you are a student from one of these colleges, the final exam must be proctored at college testing center or a Sylvan Learning Center. A valid driver's license or State ID must be shown at the testing center. An expired license or State ID will not be accepted. Use this link to help you find a college testing center or Sylvan Learning center near your home: Proctored Final Instructions

Note: Cal State Maritime, San Francisco State University, and College of the Desert provide a testing center for their students on their campus.

**Student's Requirements:**

You must have internet access.

You must have access to email.

Omega Math

^{TM}Courses - you must be at or above a 7th grade reading level. Other courses require at least a 10th grade reading level.

**Payment Plan:**

The cost of each class can be divided into two payments by checking the payment plan box when you enroll. The first payment is 55% of the total amount, and the second payment is 45% of the total amount. Your first payment will be charged when you enroll, and the second payment will be charged in 30 days. Your course is activated once your first payment is made. There is a $3.50 additional fee for the credit card split for each of the two payments.

If paying by check, you can either send in a check for the two payments 30 days apart. Or you may send in both checks at the same time, with one post dated by 30 days for the 2nd payment. Just click on "Check by Mail" under Payment Information when you enroll. The form at the end of your enrollment will give you the address for mailing the checks.

**Certificate Information:**

A certificate of completion is issued from Westcott Courses, which includes a letter grade and an overall percentage earned in the class. This course under the noncredit option does not go through UMass Global; thus, it does not carry any units and an official transcript is not included with the course. Some colleges such as Cal State Maritime, San Francisco State University, College of the Desert, etc., accept the certificate of completion for this class as a prerequisite towards another course on their campus. Please see our Associated Schools page to see if your college will accept this class under noncredit. If your college is not on this list, then they most likely want you to take the class under the credit option, which will go through UMass Global.

If you would like preapproval from your school, please see the section below on "Preapproval from your School".

**Transfer Information:**No, the noncredit version of the course is not transferable.

Courses under noncredit are not transferable. If you would like to take this class for personal enrichment, the noncredit course is the exact same class as the credit course; it is just less expensive since it is not sent through our partner university for credit. If you want to transfer the course to your college, you will need to enroll under the semester credit option. If you would like preapproval from your school, please send your counselor or registrar's office the link to this page. The noncredit courses can also be used to learn the material and then receive credit at a home college using Credit by Examination.

(K-12 use)

**Preapproval from your School:**

Westcott Courses/Omega Math is the teacher of the course, and a certificate of completion is issued. A course under the noncredit option cannot be used towards college graduation; however, it may be able to be used as a prerequisite at some colleges, such as College of the Desert, and/or graduate programs. View the Associated Colleges page to see if your school is listed. If your school is not on our list, they will most likely want the official transcript and semester credits from our credit option. Click on the link above "View the Credit Option", and then view the Preapproval tab.

## Course Information

Course Title: | Discrete Structures |

Course Code: | None |

Credits: | 0 credits |

Certificate Provider: | Westcott Courses |

Grading Mode: | Standard letter grade |

Proctored Final: | No |

**Textbook Requirement:**

No outside textbook is needed. Our Omega Math

^{TM}courses contain all the lessons, homework, solution manuals, quizzes, tests and the final. Our lessons start out with the easiest examples, and then move slowly to the more advanced problems. Between examples, there are interactive problems which make sure the student understands the concepts, as well as enables the student to store the information into long term memory.

## Course Description

This course was designed for students in math and computer science. Logic is emphasized in this course, and topics include proof and theory (including inductive and deductive proofs), propositional and predicate logic, set theory, algorithms (including recursion), trees, relations and functions, counting and probability, and elements of the theory of directed and undirected graphs (including Dijkstra's shortest path algorithm). Additionally, an introduction to complexity of algorithms and recurrence relations are included. Upon completion, students will be able to solve real world problems and use appropriate models for analysis.Discrete Structures is an Omega Math

^{TM}Course.

**Prerequisite:**College Algebra or Precalculus with a grade of C or better.

## Learning Outcomes

Upon successful completion of Discrete Structures, a student should be able to:- Effectively express themselves in a precise written form.
- Demonstrate the ability to think critically and logically.
- Students should be able to distinguish between a definition and a theorem, and apply them correctly.
- Read, comprehend and construct mathematical arguments.
- Construct standard proofs, using direct, contradiction, contrapositive, if and only if and inductive arguments.
- Demonstrate the ability to integrate knowledge and ideas in a coherent and meaningful manner.
- Represent discrete objects using the abstract mathematical structures such as sets, permutations, relations, functions, trees and graphs.
- Demonstrate appropriate techniques for specific problems in graph theory, and be able to develop and apply algorithms to solve these problems.
- Demonstrate competencies, and thought processes in the fundamental concepts of graph theory, set theory, and combinatorics, at a level necessary for more advanced mathematics courses.
- Demonstrate real-world problem solving skills. Analyze the problem and break it into parts, recognize the concepts applicable to the parts, recognize the relationship between the parts, write the concepts in proper algebraic representations, solve the problem in symbols, interpret the final results.

## Methods Of Evaluation

Homework quizzes 15%Chapter tests 60%

Final Exam 25%

(You must get at least 60% on this final in order to pass the class with a C or better.)

#### Homework Quizzes: 15%

Homework assignments are essential in a mathematics course. It is not possible to master the course without a considerable amount of time being devoted to studying the concepts and solving problems. Each lesson contains a set of homework problems, and you are required to do all the odd problems for each section. Work out each problem, and then check the solution manual for a detailed solution. Do not continue to the next problem until you understand your mistake. Once you feel comfortable with the homework set, take the homework quiz for that section. The homework quizzes are revised problems from the homework sets. You may take each quiz twice, and the higher of the two scores is used to calculate your quiz grade. Once you take a quiz, figure out what you did wrong on the problems that you missed and then try the quiz again. It is important to figure what you did wrong before you push forward. If you figure out your errors at this step, you will be less likely to make the same error on the test or the final. The struggle to figure out what you did wrong stores the mathematics into your long-term memory, and aids in building abstract thinking.#### Chapter Tests: 60%

After you have completed a chapter, and understand everything in the lessons, homework sets and quizzes, take the chapter test. The chapter tests are revised problems from the quizzes. You may take each chapter test twice, and the higher of the two scores is used to calculate your chapter test grade. Once you take a chapter test, figure out what you did wrong on the problems that you missed and then try the chapter test again. It is important to figure what you did wrong before you push forward. If you figure out your errors at this step, you will be less likely to make the same error on the final.## Assessment

A 90-100 A Clearly stands out as excellent performance and, exhibits mastery of learning outcomes.B 80-89 B Grasps subject matter at a level considered to be good to very good, and exhibits partial mastery of learning outcomes.

C 70-79 C Demonstrates a satisfactory comprehension of the subject matter, and exhibits sufficient understanding and skills to progress in continued sequential learning.

D 60-69 D Quality and quantity of work is below average and exhibits only partial understanding and skills to progress in continued sequential learning.

F 0-59 F Quality and quantity of work is below average and not sufficient to progress.

## Course Content Menu

Chapter 1 - Logic |

Lessons | Homework | Quiz |

1.1 - Nonlinear Inequalities | 1.1 | 1.1 |

1.2 - Truth Tables & Propositions | 1.2 | 1.2 |

1.3 - Logic Operators | 1.3 | 1.3 |

1.4 - Quantifiers | 1.4 | 1.4 |

1.5 - Nested Quantifiers | 1.5 | 1.5 |

Chapter 2 - Formal Proofs |

Lessons | Homework | Quiz |

2.1 - Truth Tables Proofs | 2.1 | 2.1 |

2.2 - Direct Proof | 2.2 | 2.2 |

2.3 - Proof by Contradiction | 2.3 | 2.3 |

2.4 - Proof by Contrapositive | 2.4 | 2.4 |

2.5 - If and only if Proof | 2.5 | 2.5 |

2.6 - Mathematical Induction | 2.6 | 2.6 |

Chapter 3 - Sets and Relations |

Lessons | Homework | Quiz |

3.1 - Introduction to Sets | 3.1 | 3.1 |

3.2 - Venn Diagrams | 3.2 | 3.2 |

3.3 - Binary Relations | 3.3 | 3.3 |

3.4 - Equivalence Relations | 3.4 | 3.4 |

3.5 - Matrices of Relations | 3.5 | 3.5 |

3.6 - Functions | 3.6 | 3.6 |

3.7 - Sequences & Strings | 3.7 | 3.7 |

Chapter 4 - Algorithms |

Lessons | Homework | Quiz |

4.1 - Introduction to Algorithms | 4.1 | 4.1 |

4.2 - The Euclidean Algorithm | 4.2 | 4.2 |

4.3 - Recursive Algorithms | 4.3 | 4.3 |

4.4 - Complexity of Algorithms | 4.4 | 4.4 |

Chapter 5 - Graph Theory |

Lessons | Homework | Quiz |

5.1 - Introduction to Graph Theory | 5.1 | 5.1 |

5.2 - Paths and Cycles | 5.2 | 5.2 |

5.3 - Hamiltonian Cycles | 5.3 | 5.3 |

5.4 - Matrix Representations of Graphs | 5.4 | 5.4 |

5.5 - Dijkstra's Algorithm | 5.5 | 5.5 |

5.6 - Isomorphic Graphs | 5.6 | 5.6 |

5.7 - Planar Graphs | 5.7 | 5.7 |

Chapter 6 - Trees |

Lessons | Homework | Quiz |

6.1 - Introduction to Trees | 6.1 | 6.1 |

6.2 - Depth First Search Algorithm | 6.2 | 6.2 |

6.3 - Breadth First Search Algorithm | 6.3 | 6.3 |

6.4 - Minimal Spanning Trees | 6.4 | 6.4 |

Chapter 7 - Recurrence Relations and Finite-State Machines |

Lessons | Homework | Quiz |

7.1 - Introduction to Recurrence Relations | 7.1 | 7.1 |

7.2 - Solving Recurrence Relations | 7.2 | 7.2 |

7.3 - Introduction to Finite-State Machines | 7.3 | 7.3 |

Chapter 8 - Counting and Probability |

Lessons | Homework | Quiz |

8.1 - Counting | 8.1 | 8.1 |

8.2 - Permutations and Combinations | 8.2 | 8.2 |

8.3 - Finite Basic Probabilities | 8.3 | 8.3 |

8.4 - Basic Laws of Probability | 8.4 | 8.4 |

## Time on Task

This course is online and your participation at home is imperative. A minimum of 8 - 10 hours per week of study time is required for covering all of the online material to achieve a passing grade. You must set up a regular study schedule. You have five months of access to your online account with a thirty-day extension at the end if needed. If you do not complete the course within this time line, you will need to enroll in a second term.## Schedule

Below is the suggested time table to follow to stay on a 17 week schedule for the course. The following schedule is the minimum number of sections that need to be completed each week if you would like to finish in a regular semester time frame. You do not have to adhere to this schedule. You have five months of access plus a 30 day extension at the end if needed. You can finish the course as soon as you are able, with a minimum coursework time of at least six weeks.Week | Complete Sections |

1 | 1.1 - 1.3 |

2 | 1.4 - 1.5 |

3 | 2.1 - 2.2 |

4 | 2.3 - 2.4 |

5 | 2.5 - 2.6 |

6 | 3.1 - 3.2 |

7 | 3.3 - 3.5 |

8 | 3.6 - 3.7 |

9 | 4.1 - 4.3 |

10 | 4.4 - 5.1 |

11 | 5.2 - 5.4 |

12 | 5.5 - 5.6 |

13 | 5.7 - 6.1 |

14 | 6.2 - 6.4 |

15 | 7.1 - 7.3 |

16 | 8.1 - 8.2 |

17 | 8.3 - 8.4 |

Final Exam |

## Code of Conduct:

It is the student's responsibility and duty to read the information below and become acquainted with all provisions of what constitutes academic misconduct involving cheating and plagiarism. Students are required to read each statement below, and the given repercussion. There are no exceptions to these policies, and the pretext of not reading each part will not be deemed as a reasonable excuse to contest the policies.

**Code of Ethics:**

Regulations and rules are necessary to implement for classroom as well as online course behavior. Students are expected to practice honesty, integrity and respect at all times. It is the student's responsibility and duty to become acquainted with all provisions of the code below and what constitutes misconduct.

**Respectful communications:**

When contacting Westcott Courses, you agree to be considerate and respectful. Communications from a student which are considered by our staff to be rude, insulting, disrespectful, harassing, or bullying via telephone, email, or otherwise will be considered a disrespectful communication and will result in a formal warning.

We reserve the right to refuse service. If we receive multiple disrespectful communications from person(s) representing the student, or the student themselves, the student will be excluded from taking future courses at Westcott Courses.

**Grading information and proctored final policies:**

The grading rules are put in place to protect the integrity of online education by stopping grade inflation, which is done by demanding a display of competency in exchange for a grade. By agreeing to the terms of service agreement, you agree to read the 'Grading' Policy from within your account, and the 'Proctored Final Information' page, if applicable. You have 24 hours after your first log-in to notify us if you do not agree to the grading policy and proctored final policy ( if applicable ) outlined in the pages inside of your account, otherwise it is assumed that you agree with the policies. There are no exceptions to these policies, and the pretext of not reading the pages will not be deemed as a reasonable excuse to contest the policies.

The definition of academic cheating is an act of dishonesty in order to obtain a higher grade in the course, and to gain an advantage over other students in the course.To maintain academic standards, students are expected to practice honesty, integrity and respect at all times. Students who violate the policies of cheating, plagiarizing, or other academic misconduct will result in following actions.

1) Cheating in any way on the final exam results in an F on the final and an F in the class. This includes, but is not limited to any form of collaboration, use of unauthorized materials, receiving or providing unpermitted assistance on the exam, using outside digital assistance such as a cell phone, tablet, ETC. to communicate with others or access outside websites, having someone else take the exam for you, taking an exam for another student, failing to stop working on the exam when the time is up.

Final exams are secure tests and the intellectual property of Westcott Courses. Taking screen shots of a digital final or copying a paper test is stealing our intellectual property and cheating. It is equivalent to stealing a copy of the final exam off an instructor's desk. When one student obtains the questions on a final, it means that other students who don't have the questions on the final are at a disadvantage. Once a final exam has been compromised it is no longer secure, and the exam is unfair for those who have not performed an act of dishonesty to gain the advantage.

Each of the infractions above represents a result of performing an act of dishonesty in order to obtain a higher grade in the course, and to gain an advantage over other students in the course. The result of any of the above offenses is an F in the course. Students who violate the above policy may retake the course after a first offense; however, a second offense will result in expulsion and students will no longer be able to take other courses at Westcott Courses.

Students are responsible for clicking on the “Proctored Final Information” link (which is on student’s Main Menu), and reviewing the list of Authorized Materials for each course's final exam. Since each course is different, the “Authorized Materials” for each final is different. For example, some courses permit notes, while others do not.

2) Plagiarism: All of the following are considered plagiarism, and will result in a zero on the plagiarized assignment, and there are no opportunities to redo the assignment.

Merriam-Webster defines plagiarism as “the act of using another person's words or ideas without giving credit to that person” Plagiarism includes, but is not limited to:

- having somebody else write your assignment for you
- turning in an assignment that contains work that is not your own
- changing words in phrases, sentences and/or blocks of text without giving credit to the source (paraphrase)
- copying ideas, phrases, sentences or entire blocks of text without giving credit to the source
- not crediting the correct source by providing incorrect information

Plagiarism is an act of fraud, and can usually be avoided by using quotation marks and citing the source of the material. Instructors apply plagiarism software to find assignments that contain plagiarized material. Again, assignments that contain one of the above infractions will receive a zero on the assignment and the student will not have the opportunity to redo the assignment.

It is important to note that saving all your assignments to the end of the course, and then turning in multiple assignments that have been plagiarized will result in zeros on all of those assignments. This may mean that you no long have enough points in the course to pass the class. Thus, turning in assignments one at a time and waiting for instructor feedback in-between is important for learning and making sure that you maximize your possible points.

If you have questions, please read more information about plagiarism at plagiarism.org, or ask your instructor.

**Other Examples of Academic Misconduct:**

1) Other forms of cheating include altering an exam and submitting it for regarding, providing false excuses to postpone due dates, fabricating data or references, claiming that Westcott Courses lost your test and/or quiz scores, sending emails to Westcott claiming you did not know what you were doing was cheating.

2) Unauthorized collaboration - working with others on graded course work without specific permission of the instructor, including homework assignments, programs, quizzes and tests.

3) Copying Westcott Courses content and posting it on the internet. This includes assignments, quizzes, and tests.

By signing up for a course, you are legally signing a contract that states that the person who is named taking this course is the actual individual doing the course work and all examinations. You also agree that for courses that require proctored testing, that your final will be taken at a college testing center, a Sylvan Learning center, or at home using the online proctor. Also, the individual signed up for this course will be the one taking the test. Failure to do so will be considered a breach of Westcott Courses policies.

**Important Notes:**

This syllabus is subject to change and / or revision during the academic year. Students with documented learning disabilities should notify our office upon enrollment, as well as make sure we let the testing center know extended time is permitted. Valid documentation involves educational testing and a diagnosis from a college, licensed clinical psychologist or psychiatrist.

#### Related Information:

#### How it Works

These courses are available to anyone in the public who'd like to enroll. Unlike traditional colleges where one must enroll in a degree program, Westcott Courses offers individual courses that are available instantly. Westcott Courses is the teacher of the course, and the official transcript with academic semester credits comes from our partner university, UMass Global who is regionally accredited by WASC. Westcott Courses doesn't appear on the transcript; the transcript shows our courses the same way it would show a traditional class taken at the university's campus.Read more about transferring the credits.

#### Teachers for the course:

The classes are self-paced; however, a teacher is attached to each course who holds at least a master's degree in their field of study (or related). Most of our teachers are teachers who teach the exact same course on either a California State University or Community college campus.#### Omega Math^{TM} Courses:

Courses that have the trademark of "Omega Math^{TM}" are courses that have been written by our Omega Math team, and have a format that students love! These courses contain all the lessons, homework, solution manuals, quizzes, tests and the final. Students can take each test and quiz twice, and only the higher of the two scores counts towards their grade. This feature enables students to figure out what they did wrong before pushing forward, providing them with a deeper understanding of the course material.