# Methods of Proof in Mathematics

### Methods of Proof in Mathematics Syllabus

• Course Code: MATU 220

Transcript:

Yes. Your transcript will come from the records office at Brandman University. They are regionally accredited and award semester credits.

Credits: 4 semester

Transfer: 4 year degree applicable

Enrollment Schedule:

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 4 - 8 weeks. The upper level math classes, such as Calculus and above, usually take students 3-4 months. (Note: The 30-day extension cannot take your total course time six months beyond the date of enrollment. At the end of the six months, we must post a grade with the university.)

Required Textbook:

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

Proctored Final: Yes

#### 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.
Prerequisite: Calculus l, Calculus ll, and Calculus lll

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor. An official syllabus will be provided to the student on the first day of class by the instructor.

#### Learning Outcomes

At the conclusion of this course, students should be able to:

1. Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments.
2. Perform set operations on finite and infinite collections of sets.
3. Determine equivalence relations on sets and equivalence classes.
4. Identify functions, surjections, injections, and bijections and work with inverse images and inverse functions.
5. Apply multiple techniques of mathematical proof (direct, contradiction, conditional, contraposition, and induction).
6. 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 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.

Proctored Final

The final exam is a comprehensive final covering all of the chapters.
No notes are permitted on the final. Scratch paper, pencil and eraser are the only materials permitted on the final.

The final exam must be proctored at college testing center or a Sylvan Learning Center. The final may only be taken within the U.S. A valid driver's license must be shown at the testing center. An expired license will not be accepted.

The 60% rule was set in place to protect the integrity of online math education by requiring a display of competency in exchange for a grade. All schools which are regionally accredited adhere to online standards. Your college is accepting this course because it goes through a regionally accredited university, which tells your college that standards have been met. Your college will not accept a class from a school that is not regionally accredited, because they know the standards won't be met.

#### 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.

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

2.8. 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

5.3. 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

## Conduct Code:

#### 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:

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.

• Course Code: None

Transcript:

A certificate of completion is issued by Westcott Courses. If this course is taken under the non-credit option, it does NOT go through our partner university, Brandman University. Thus, a transcript is not included with the course. A certificate of completion is awarded for students who receive a C or better in the course.

Credits: 0

Certificate of Completion: Yes

Transfer:

If you would like to take this class for personal enrichment, the non-credit 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 pre-approval from your school, please send your counselor or registrar's office the link to this page. The non-credit courses can also be used to learn the material and then receive credit at a home college using Credit by Examination. (K-12 use)

Enrollment Schedule:

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 4 - 8 weeks. The upper level math classes, such as Calculus and above, usually take students 3-4 months. (Note: The 30-day extension cannot take your total course time six months beyond the date of enrollment. At the end of the six months, we must post a grade with the university.)

Required Textbook:

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

Proctored Final: No

#### 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.
Prerequisite: Calculus l, Calculus ll, and Calculus lll

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor. An official syllabus will be provided to the student on the first day of class by the instructor.

#### Learning Outcomes

At the conclusion of this course, students should be able to:

1. Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments.
2. Perform set operations on finite and infinite collections of sets.
3. Determine equivalence relations on sets and equivalence classes.
4. Identify functions, surjections, injections, and bijections and work with inverse images and inverse functions.
5. Apply multiple techniques of mathematical proof (direct, contradiction, conditional, contraposition, and induction).
6. 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 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:
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.

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

2.8. 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

5.3. 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

## Conduct Code:

#### 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:

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.