# Methods of Proof in Mathematics

**Course Code:**None

**Credits:**Noncredit

**Certificate Via Westcott**

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**Credit Information**

Course Title: | Methods of Proof in Mathematics |

Course Code: | None |

Credits: | Noncredit |

Certificate Provider: | Westcott Courses |

Transfer: | No, but can be used at some colleges as a prerequisite. |

Grading Mode: | Standard letter grade |

**Course Requirements**

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

Requires Textbook? | Yes | This course requires a textbook. |

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

**Course Description**

This course is an introduction to abstract mathematics, with an emphasis on the techniques of mathematical proof (direct, contradiction, conditional, contraposition). Topics to be covered include logic, set theory, relations, functions and cardinality.

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor.

**Prerequisite:**Calculus l, Calculus ll and Calculus lll (Or Business Calculus, Calculus ll and Calculus lll.)

**Course Pricing**

The full cost of the course is $693.00; however, you may use the payment plan to split the cost into two payments.

**Payment Plan:**

1st payment: $384.65 (Start your class immediately!)

2nd payment: $315.35 (Automatically charged 30 days later.)

Enroll Now - Noncredit - $693 View Credit Option - $973

**Helpful Links:**

Math Placement Test

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. Most students finish the lower level courses in 1 - 3 months. The upper level math courses, such as Calculus and above, usually take students 2 - 4 months.

**Textbook Requirement:**

Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition)

- Chartrand, Gary; Polimeni, Albert; Zhang, Ping

- Textbook ISBN-10: 0321797094

- Textbook ISBN-13: 978-0321797094

Text Book on Amazon

Check the price on abebooks.com, as their price is usually lower than Amazon.**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 Brandman University; 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 Brandman University.

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

**Transfer Information:**No, but can be used at some colleges as a prerequisite.

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: | Methods of Proof in Mathematics |

Course Code: | None |

Credits: | 0 credits |

Certificate Provider: | Westcott Courses |

Grading Mode: | Standard letter grade |

Proctored Final: | No |

**Textbook Requirement:**

Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition)

- Chartrand, Gary; Polimeni, Albert; Zhang, Ping

- Textbook ISBN-10: 0321797094

- Textbook ISBN-13: 978-0321797094

Text Book on Amazon

Check the price on abebooks.com, as their price is usually lower than Amazon.

## Course Description

This course is an introduction to abstract mathematics, with an emphasis on the techniques of mathematical proof (direct, contradiction, conditional, contraposition). Topics to be covered include logic, set theory, relations, functions and cardinality.Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor.

**Prerequisite:**Calculus l, Calculus ll and Calculus lll (Or Business Calculus, Calculus ll and Calculus lll.)

## Learning Outcomes

At the conclusion of this course, students should be able to:- Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments.
- Perform set operations on finite and infinite collections of sets.
- Determine equivalence relations on sets and equivalence classes.
- Identify functions, surjections, injections, and bijections and work with inverse images and inverse functions.
- Apply multiple techniques of mathematical proof (direct, contradiction, conditional, contraposition, and induction).
- Write solutions to problems and proofs of theorems that meet rigorous standards based on content, organization and coherence.

## Methods Of Evaluation

Homework 15%Exams (4) 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:**

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 using the concepts to solve problems. Each lesson contains a set of homework problems.

**Exams (4):**

The exams are designed to cover a broader area of the text and test your understanding of the material.

## 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.

## Content Menu

#### Course Content Menu:

**Chapter 1. Sets**

1.1. Describing a Set

1.2. Subsets

1.3. Set Operations

1.4. Indexed Collections of Sets

1.5. Partitions of Sets

1.6. Cartesian Products of Sets

**Chapter 2. Logic**

2.1. Statements

2.2. The Negation of a Statement

2.3. The Disjunction and Conjunction of Statements

2.4. The Implication

2.5. More on Implications

2.6. The Biconditional

2.7. Tautologies and Contradictions

2.8. Logical Equivalence

2.9. Some Fundamental Properties of Logical Equivalence

2.10. Quantified Statements

2.11. Characterizations of Statements

**Chapter 3. Direct Proof and Proof by Contraposition**

3.1. Trivial and Vacuous Proofs

3.2. Direct Proofs

3.3. Proof by Contrapositive

3.4. Proof by Cases

3.5. Proof Evaluations

**Chapter 4. More on Direct Proof and Proof by Contrapositive**

4.1. Proofs Involving Divisibility of Integers

4.2. Proofs Involving Congruence of Integers

4.3. Proofs Involving Real Numbers

4.4. Proofs Involving Sets

4.5. Fundamental Properties of Set Operations

4.6. Proofs Involving Cartesian Products of Sets

**Chapter 5. Existence and Proof by Contradiction**

5.1. Counterexamples

5.2. Proof by Contradiction

5.3. A Review of Three Proof Techniques

5.4. Existence Proofs

5.5. Disproving Existence Statements

**Chapter 6. Mathematical Induction **

6.1. The Principle of Mathematical Induction

6.2. A More General Principle of Mathematical Induction

6.3. Proof By Minimum Counterexample

6.4. The Strong Principle of Mathematical Induction

**Chapter 8. Equivalence Relations**

8.1. Relations

8.2. Properties of Relations

8.3. Equivalence Relations

8.4. Properties of Equivalence Classes

8.5. Congruence Modulo n

8.6. The Integers Modulo n

**Chapter 9. Functions**

9.1. The Definition of a Function

9.2. The Set of All Functions from A to B

9.3. One-to-one and Onto Functions

9.4. Bijective Functions

9.5. Composition of Functions

9.6. Inverse Functions

9.7. Permutations

**Chapter 10. Cardinalities of Sets**

10.1. Numerically Equivalent Sets

10.2. Denumerable Sets

10.3. Uncountable Sets

Exam #4

**Final Exam for Methods of Proof**

## 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.## Code of Conduct:

**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. Cheating is forbidden of any form will result in an F in the class.

**Respectful communications:**

When contacting Omega Math or 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/Omega Math.

**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.

**Examples of academic misconduct:**

Cheating: Any form of cheating will result in an F in the class. If there is an associated college attached to the course, that college will be notified of the F due to cheating and they will determine any disciplinary action.

Any form of collaboration or use of unauthorized materials during a quiz or an exam is forbidden.

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, and the individual signed up for this course will be the one taking the test. Failure to do so will be considered a breach of contract.

Other forms of cheating include receiving or providing un-permitted assistance on an exam or quiz; taking an exam for another student; using unauthorized materials during an exam; altering an exam and submitting it for re-grading; failing to stop working on the exam when the time is up; providing false excuses to postpone due dates; fabricating data or references, claiming that Westcott Courses/Omega Math lost your test and or quiz scores. This includes hiring someone to take the tests and quizzes for you.

**Unauthorized collaboration:**

Working with others on graded course work without specific permission of the instructor, including homework assignments, programs, quizzes and tests, is considered a form of cheating.

**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 individual 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, Brandman University who is regionally accredited by WASC. Westcott Courses does not 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 California Community college campus.#### Omega Math^{TM} Courses:

Courses that have the trademark of "Omega Math^{TM}Courses", 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.