Download Catalog Description A study of proof techniques used in mathematics

Exploration, Proof, and Writing in Mathematics
Ted Sundstrom, Grand Valley State University
[email protected]
http://faculty.gvsu.edu/sundstrt/
Math Fest, August 2002
MTH 210 – Communicating in Mathematics A study of proof techniques used in
mathematics. Intensive practice in reading mathematics, expository writing in mathematics, and
constructing and writing mathematical proofs. Mathematical content will be selected from the
areas of logic, set theory, number theory, relations and functions. Prerequisite: Calculus I
Required Content: Logic, Quantifiers with an emphasis on negations, elementary set theory,
functions (including injections, surjections, and inverse functions), number theory (including
congruence arithmetic), equivalence relations.
Supplemental Writing Skills Requirements
MTH 210 is always offered as a Supplemental Writing Skills (SWS) Course. Following are the
university requirements for an SWS course:
Completion of English with a grade of C or better (not C-) is a prerequisite. SWS credit
will not be given to a student who completes this course before completing English 150.
SWS courses adhere to certain guidelines. Students turn it a total of at least 3000 words
of writing during the semester. Part of that total may be essay exams, but a substantial
amount of it is made up of finished essays or reports or research papers. The instructor
works with the students on revising drafts of their papers, rather than simply grading the
finished pieces of writing. At least four hours of class time are devoted to writing
instruction. At least one-third of the final grade is based on the writing assignments.
SWS credit will be given only if a student completes this course with a grade of C or better.
Course Objectives
1. To develop the ability to read and understand written mathematical proofs.
2. To develop the ability to construct and write mathematical proofs using standard methods of
mathematical proof including direct proofs, mathematical induction, case analysis, and
counterexamples.
3. To develop logical thinking skills and to develop the ability to think more abstractly in a
proof oriented setting.
4. To develop talents for creative thinking and problem solving.
5. To improve the quality of communication in mathematics. This includes improving writing
techniques, reading comprehension, and oral communication in mathematics.
6. To explore and understand the concepts described in the course content above.
MTH 210 - Winter 2001
Prof. Sundstrom
Guidelines for The Portfolio Problems
Following are the ten Portfolio Problems for the Portfolio Project for this course. The Portfolio
will be worth a total of 120 points. Each problem will be worth 10 points, and there will be 20
points possible for submission of proofs for review by the professor. To be eligible for the 20
points, a student must do all of the following:
•
•
•
Submit the first drafts of a portfolio problem by Monday February 11, 2002;
Submit the first draft of a second portfolio problem (different from the first) by Monday
February 18, 2002; and
Submit the first draft of a third portfolio problem (different than the first two) by Monday
February 25, 2002.
In addition, there will be ten “bonus points” available for the Portfolio Project. These bonus
points will be awarded to each student who has received a score of 10 on three portfolio
problems by Wednesday March 27, 2002.
Following are some important guidelines and rules for the Portfolio Project:
•
•
•
•
•
You may hand in a given Portfolio Problem to the professor two times to be critiqued.
No more than two Portfolio Problems may be submitted for review on a given day.
No more than four Portfolio Problems may be submitted for review during any given week.
The last date to have a Portfolio Problem critiqued is Wednesday April 10, 2002.
The final Portfolio is due no later than the beginning of class on Wednesday April 17, 2001.
Please refer to the document "Guidelines for the Portfolio Theorems" for details regarding the
requirements for the Portfolio Theorems.
Following is a description of the 10-point scale for grading each theorem:
Points
10
9
6
3
0
Description
The solution is correct and well written according to the
writing guidelines for the course.
The solution is correct but there is a writing mistake.
The solution is essentially correct but the solution is not
written according to the guidelines for the course.
Significant progress has been made in developing and
writing a solution for the problem.
Little or no progress has been made in developing a
solution for the problem.
MTH 210
Prof. Sundstrom
The Portfolio Problems
Problem One
Find all solutions of two quadratic equations of the form ax 2 + bx + c = 0 where a, b, and c are
real numbers, a > 0 , and c < 0 .
Prove or disprove the following:
If a, b, and c are real numbers with a > 0 and c < 0 , then at least one solution of the
quadratic equation ax 2 + bx + c = 0 is a positive real number.
Problem Two
One of the most famous unsolved problems in mathematics is a conjecture made by Christian
Goldbach in a letter to Leonhard Euler in 1742. The conjecture made in this letter is now known
as Goldbach’s Conjecture.
State Goldbach’s Conjecture and explain what it would take to prove that Goldbach’s conjecture
is false. Then, prove the following:
If there exists an odd integer greater than 5 that is not the sum of three prime numbers, then
Goldbach’s Conjecture is false.
Problem Three
Is the real number
12 a rational number or an irrational number? Justify your conclusion.
Problem Four
Prove or disprove the following:
There exist three consecutive natural numbers such that the cube of the largest one is equal to
the sum of the cubes of the other two.
Problem Five
If n is a natural number and m = n + 1 , then n and m are said to be consecutive natural
numbers. If n is an odd natural number and m = n + 2 , then n and m are said to be
consecutive odd natural numbers.
Notice that 3, 5, and 7 are three consecutive odd natural numbers, all of which are prime. Are
there any others? Either find three other consecutive odd natural numbers, all of which are
prime, or prove that, except for 3, 5, and 7, every triple of consecutive odd natural numbers
contains at least one composite number.
MTH 210
Prof. Sundstrom
The Portfolio Problems
Problem Six
Prove or disprove the following:
For any sets A, B, and C that are subsets of a universal set U,
A− B∩C = A− B ∪ A−C .
b
g b
g b
g
Problem Seven
Let n be a natural number with n ≥ 3 . A convex polygon with n sides is a polygon with n
sides with the additional property that the straight line segment between any two points of the
polygon lies entirely within the polygon. So, for example, a triangle is a convex polygon with 3
sides. In Euclidean geometry, what is the sum of the interior angles, in radians, of a triangle?
Develop a formula for the sum of the interior angles, in radians, of the interior angles of a convex
polygon with n sides and prove that your formula is correct.
Problem Eight
Prove or disprove the following:
Let f 1 , f 2 , f 3 ,… , f m ,… be the sequence of Fibonacci numbers. Then, for all natural numbers
n, f 5n is a multiple of 5.
Problem Nine
Give examples of three different pairs of prime numbers that differ by two. Such pairs of
numbers are said to be twin primes. Calculate the product of each of your examples of twin
primes. Is the following proposition true or false:
If m and n are twin primes other than the pair 3 and 5, then mn + 1 is a perfect square that
is divisible by 36.
Problem Ten
Let A, B, and C be sets, and let f : A → B and g : B → C be functions. If the composite
function g f : A → C is a injection, then the function f : A → B is a injection.
Prove or disprove each of the following:
• If the composite function g f : A → C is a injection, then the function f : A → B is a
injection.
•
If the composite function g f : A → C is a injection, then the function g : B → C is a
injection.
MTH 210
Prof. Sundstrom
Guidelines for Writing Mathematical Proofs
From: Mathematical Reasoning: Writing and Proof, by Ted Sundstrom, © 2003 Prentice Hall.
One of the most important forms of mathematical writing is writing mathematical proofs. The
writing of mathematical proofs is an acquired skill and takes a lot of practice. Throughout the
textbook, we have introduced various guidelines for writing proofs. Following is a summary of
all the writing guidelines introduced in the text. This summary contains some standard
conventions that are usually followed when writing a mathematical proof.
1. Begin with a carefully worded statement of the theorem or result to be proven. The
statement should be a simple declarative statement of the problem. Do not simply rewrite the
problem as stated in the textbook or given on a handout. Problems often begin with phrases
such as “Show that” or “Prove that.” This should be reworded as a simple declarative
statement of the theorem. Then skip a line and write “Proof” in boldface font (when using a
word processor). Begin the proof on the same line. Make sure that all paragraphs can be
easily identified. Skipping a line between paragraphs or indenting each paragraph can
accomplish this.
As an example, an exercise in a text might read, “Prove that if x is an odd integer, then x 2
is an odd integer.” This could be started as follows:
Theorem. If x is an odd integer, then x 2 is an odd integer.
Proof: We assume that x is an odd integer . . .
2. Begin the proof with a statement of the assumptions. This is illustrated in the example in
Part (1). Follow the statement of the assumptions with a statement of what will be proven.
Theorem. If x is an odd integer, then x 2 is an odd integer.
Proof: We assume that x is an odd integer, and we will prove that x 2 is an odd integer.
3. Use the pronoun “we.” If a pronoun is used in a proof, the usual convention is to use “we”
instead of “I.” The idea is that the author and the reader are proving the theorem together.
4. Use italics for variables. When using a word processor to write mathematics, the word
processor needs to be capable of producing the appropriate mathematical symbols and
equations. The mathematics that is written with a word processor should look like typeset
mathematics. This means that variables need to be italicized, boldface is used for vectors,
and regular font is used for mathematical terms such as the names of the trigonometric
functions and logarithmic functions. The use of italics is illustrated in the example in
Part (2).
5. Use complete sentences and proper paragraph structure. Good grammar is an important
part of any writing. Therefore, conform to the accepted rules of grammar. Pay careful
attention to the structure of sentences. Write proofs using complete sentences but avoid runon sentences. Part of good grammar is correct spelling. Always use a spell checker when
using a word processor.
MTH 210
Prof. Sundstrom
Guidelines for Writing Mathematical Proofs
6. Keep the reader informed. Sometimes a theorem is proven by proving the contrapositive
or by using a proof by contradiction. If either proof method is used, this should be indicated
within the first few lines of the proof. This also applies if the result is going to be proven
using mathematical induction.
Examples:
We will prove this result by proving the contrapositive of the statement.
•
•
We will prove this statement using a proof by contradiction.
•
We will assume to the contrary that . . .
•
We will use mathematical induction to prove this result.
In addition, make sure the reader knows the status of every assertion that is made. That is,
make sure it is clearly stated whether an assertion is an assumption of the theorem, a
previously proven result, a well-known result, or something from the reader's mathematical
background.
7. Display important equations and mathematical expressions. Equations and manipulations
are often an integral part of the exposition. Do not write equations, algebraic manipulations,
or formulas in one column with reasons given in another column (as is often done in
geometry texts). Important equations and manipulations should be displayed. This means
that they should be centered with blank lines before and after the equation or manipulations.
If several steps are shown together, the equals signs should be aligned, and if one side of an
equation does not change, it should not be repeated. For example,
Using algebra, we obtain
x ⋅ y = (2m + 1)(2n + 1)
= 4mn + 2m + 2n + 1
= 2 (2mn + m + n ) + 1
Since m and n are integers, we conclude that . . .
8. Equation numbering guidelines. If it is necessary to refer to an equation later in a proof,
that equation should be centered and displayed, and it should be given a number. The
number for the equation should be written in parentheses on the same line as the equation at
the right-hand margin.
Example: Since x is an odd integer, there exists an integer n such that
x = 2n + 1 .
(1)
MTH 210
Prof. Sundstrom
Guidelines for Writing Mathematical Proofs
Later in the proof, there may be a line such as
Then, using the result in Equation (1), we obtain . . .
Please note that we should only number those equations we will be referring to later in the
proof. Also, note that the word “Equation” begins with a capital “E” when we are referring
to an equation by number.
9. Do not begin a sentence with a mathematical symbol. In addition to not beginning a
sentence with a mathematical symbol, in formal writing in mathematics, we do not use the
special symbols ∀ (for all), ∃ (there exists), ∋ (such that), or ∴ (therefore).
10. Tell the reader when the proof has been completed. Perhaps the best way to do this is to
say outright that, “This completes the proof.” Although it may seem repetitive, a good
alternative is to finish a proof with a sentence that states precisely what has been proven. In
any case, it is usually good practice to use some “end of proof symbol” such as „.
11. Write a first draft of your proof and then revise it. Remember that a proof is written so
that readers are able to read and understand the reasoning in the proof. Be clear and concise.
Include details but do not ramble. Do not be satisfied with the first draft of a proof. Read it
over and refine it. Just like any worthwhile activity, learning to write mathematics well takes
practice and hard work. This can be frustrating. Everyone can be sure that there will be
some proofs that are difficult to construct, but remember that proofs are a very important part
of mathematics. So work hard and have fun.
MTH 210
Prof. Sundstrom
Some Preview Activities
From: Mathematical Reasoning: Writing and Proof, by Ted Sundstrom, © 2003, Prentice Hall.
Definition of Divides, Divisor, Multiple
Definition
An integer m divides an integer n provided that there is an integer q such that n = m ⋅ q . We
also say that m is a divisor of n, m is a factor of n, and that n is a multiple of m.
This definition can be written in symbolic form using appropriate quantifiers as follows:
An integer m divides an integer n provided that ( ∃q ∈ Z ) ( n = m ⋅ q ) .
1. Give three different examples of three integers where the first integer divides the second
integer and the second integer divides the third integer.
2. In your examples in Part (1), is there any relationship between the first and the third integer?
Explain, and formulate a conjecture. Write your conjecture in the form of a conditional
statement.
3. Give several examples of two integers where the first integer does not divide the second
integer.
4. According to the definition given above, does the integer 0 divide the integer 10? That is, is
0 a divisor of 10? Explain.
5. According to the definition given above, does the integer 10 divide the integer 0? That is,
is 10 a divisor of 0? Explain.
6. Use the definition given above to complete the following sentence in symbolic form: The
integer m does not divide the integer n means that … .
7. Use the definition given above to complete the following sentence without using the symbols
for quantifiers: “The integer m does not divide the integer n . . . ”
Calendars and Clocks (Preview for Congruence)
1. Suppose that it is currently Tuesday.
a. What day will it be 3 days from now?
b. What day will it be 10 days from now?
c. What day will it be 17 days from now? What day will it be 24 days from now?
d. Find several other natural numbers x such that it will be Friday x days from now.
e. Create a list (in increasing order) of the numbers 3, 10, 17, 24, and the numbers you
generated in Part (d). Pick any two numbers from this list and subtract one from the
other. Repeat this several times.
f. What do the numbers you obtained in Part (e) have in common?
2. Suppose that we are using a twelve-hour clock with no distinction between A.M. and P.M.
Also, suppose that the current time is 5:00.
a. What time will it be 4 hours from now?
b. What time will it be 16 hours from now? What time will it be 28 hours from now?
c. Find several other natural numbers x such that it will be 9:00 x hours from now.
MTH 210
Prof. Sundstrom
Some Preview Activities
d. Create a list (in increasing order) of the numbers 4, 16, 28, and the numbers you
generated in Part (c). Pick any two numbers from this list and subtract one from the
other. Repeat this several times.
e. What do the numbers you obtained in Part (d) have in common?
3. This is a continuation of Part (1). Suppose that it is currently Tuesday.
a. What day was it 4 days ago?
b. What day was it 11 days ago? What day was it 18 days ago?
c. Find several other natural numbers x such that it was Friday x days ago.
d. Create a list (in increasing order) consisting of the numbers -18, -11, -4, the negatives of
the numbers you generated in Part (c) and the positive numbers in the list from
Part (1, e). Pick any two numbers from this list and subtract one from the other. Repeat this
several times.
e. What do the numbers you obtained in Part (d) have in common?
The Sum of the Divisors Function (Introduction to Functions)
Let s be the function that associates with each natural number the sum of its distinct natural
number factors. For example,
s (6 ) = 1 + 2 + 3 + 6
= 12.
1. What is the domain of the function s ?
2. Calculate s ( k ) for each natural number k from 1 through 15.
3. Is s
( 5)
defined? Explain. Is s (π ) defined? Is s ( −6 ) defined?
4. If n ∈ N , write a sentence describing how to calculate s ( n ) .
5. Does there exist a natural number n such that s ( n ) = 5 ? Justify your conclusion.
6. Is it possible to find two different natural numbers m and n such that s ( m ) = s ( n ) ?
Explain.
7. Are the following statements true or false?
a. For each m ∈ N , there exists a natural number n such that s ( n ) = m .
b. If m, n ∈ N and m ≠ n , then s ( m ) ≠ s ( n ) .