Introductory Proofs

“Mastering Mathematical Concepts and Proof Techniques: A Comprehensive Portfolio for MTH 299, Spring 2024” Introduction to Proof Techniques and Set Operations in Mathematics

MTH 299 Portfolio: Spring 2024
The portfolio is meant to help you create a big picture view of the course by summarizing the main proof
techniques and important mathematical concepts covered in this class. We also hope that your portfolio will
serve as a valuable reference beyond this course.
General Guidelines
Please follow the instructions below.
• The portfolio should be completed individually, but you are encouraged to ask your instructors and TAs
for help and guidance.
• It will consist of 5 sections, as described on the next pages.
• A draft (worth 10% of the total portfolio grade), including the first four sections of the portfolio is due
on Gradescope on Friday, March 22th, 2024. The draft will not be graded for correctness, but you
will receive feedback to help improve your final portfolio.
• The complete portfolio (worth 90% of the total portfolio grade) is due on Gradescope on Friday, April
12th, 2024. The complete portfolio will be graded for correctness, level of explanation, style/writing
quality, and originality.
• The portfolio should be professional and needs to be typed in LaTeX. LaTeX resources are available on
D2L. We will also have a LaTeX tutorial in class on Thursday, February 22.
• Imagine you are writing a textbook. Write in a way which will be helpful to you now, and in the future.
Ask yourself: “If I look at this a year from now will I understand it?” if not, try rewording things.
• Make sure your portfolio is well organized so that you and anyone else can read it easily.
• Write in a concise way, you won’t want to sift through excess information.
• Make sure that what you write is correct!
Use of artificial intelligence is not permitted on your portfolio.
MTH299 – Portfolio, Spring 2024
Section Instructions: Your portfolio should consist of 5 sections. Below you will find instructions about what
to include in each section. In some parts of the sections, we ask you to come up with examples. In each
such part, to clarify the level of originality we expect from your examples, we will provide one of two labels:
“Original” or “Reference”. The labels mean the following:
• Original: This means you must come up with your own example(s) and not use an example from class,
the course materials/textbook, or outside resource.
• Reference: This means you are allowed to use an example from class, the course materials/textbook,
or outside reference. In this situation, next to your example, you should reference its source. If it
is an example from class, write “Source: Class material”. If it is from the textbook, reference the
example/exercise number and page number. If it is from a homework, worksheet, or quiz, reference
the worksheet/homework/quiz number and the problem number. If it is from an online article, provide
the url. If it is from another outside resource like a book other than our textbook, provide the title of
the book and the page number. If a section part has the label “Reference”, you are allowed to come
up with an original example instead, and in fact we encourage it. Note: any referenced examples
should not be taken from the materials we provide in the portfolio folder on D2L.
1 Statements, Implications, and Quantifiers
(a) Provide two different examples of sentences that are statements. Also, provide two different examples
of sentences that are not statements. For each of your four total examples, explain why or why not it
is a statement. Provide advice to the reader on how to detect if a sentence is a statement. Original
(b) Explain the misunderstanding the following person has about the definition of a statement.
Person: The sentence “The 299 quadrillionth digit of π is equal to 3.” is not a statement because I don’t
know whether or not it is true.
(c) Provide an example of an implication P =⇒ Q such that P =⇒ Q is true, but Q =⇒ P is false.
Explain. Also, provide a different example of an implication R =⇒ S such that R =⇒ S and
S =⇒ R are both true. Explain. Original
(d) Describe the meaning of the quantifiers ∀, ∃. Explain how to negate statements involving these quanti-
fiers. Provide advice to the reader on how to avoid mistakes when working with ∀, ∃.
2 Direct Proof
(a) Outline the general steps involved in directly proving an implication, P =⇒ Q. Come up with an
implication and prove it using direct proof. Original
(b) Give an example of a statement that is not an implication and prove it using direct proof. Reference
(c) When do you think direct proof is the better way to prove a statement, rather than using an indirect
proof like proof by contrapositive or contradiction.
(d) What are some common mistakes and logical fallacies that can occur while using the method of direct
MTH299 – Portfolio, Spring 2024
3 Proof by Contradiction and Contrapositive
(a) List the general steps of a proof by contrapositive, and provide a few examples of the types of state-
ments/theorems that are best suited for this proof technique. Provide a proof of one of these state-
ments/theorems, identifying each step as you described them above. Original
(b) Outline the differences between a proof by contrapositive and a proof by contradiction, and also what
they have in common. Identify how one might confuse the two techniques, and how to avoid this.
(c) Provide an example of a statement/theorem that is best suited for contradiction, as opposed to contra-
positive. Explain why proof by contradiction is the better choice for proving this statement. Include a
proof by contradiction of this statement. Reference
4 Proof by Induction
(a) Provide an outline for a proof by induction, including each of its main components. Explain why proof
by induction works, and what types of statements can typically be proven.
(b) Provide two different examples of statements/theorems that can be proven using induction and include
proofs. One example must use the principle of strong induction. Reference
(c) Explain at least two common mistakes that can be made when writing a proof by induction and how
someone writing a proof can avoid these mistakes.
5 More Mathematics
(a) Describe the set operations: union, intersection, and difference. Provide specific examples to explain
how these operations work. Original
(b) Explain the definition of a function. Provide advice to the reader on how to avoid common misconcep-
tions about functions.
(c) Describe how to perform the Euclidean algorithm via an example. In other words, pick two integers and
perform the algorithm to find their GCD. Annotate your example to explain to the reader your steps.
You should pick your pair of integers so that there are at least 5 steps in your algorithm. Do not pick
something trivial like the pair 1, 2. Original
(d) Explain some of the subtleties of the concept of cardinality, specifically in the context of infinite sets.
Provide advice to the reader on how to think about the cardinality of infinite sets and how to avoid
common mistakes.