Course Content and
Goals:
This course serves as a
stepping stone between the lower level math courses and the upper
level math courses. As you progress in the study of mathematics,
it becomes necessary to learn the theory of mathematics, which requires
being able to understand the proofs of mathematical theorems and
to prove results yourself. In this course, you will learn about logic,
proof techniques, and basic structures of mathematics. You will apply
these new tools to prove theorems and solve problems in many areas of
mathematics. The theorems and problems serve as motivation for learning
proof techniques and come from a wide range of topics including number
theory, abstract algebra, discrete math, and continuous math. The tools
used to prove theorems and understand mathematical theory include quantifiers,
mathematical statements, truth tables, equivalence of statements,
venn diagrams, induction, the Well-Ordering Property, direct proof,
proof by contradiction, permutations, arrangements, selections, and
partitions. We will also cover some of the basic structures of mathematics
including, field axioms, ordered field axioms, number systems, sets,
cartesian product, n-tuples, q-ary representations, basic set theory,
functions, cardinality, and combinatorial graphs.
While learning the skills of proving theorems, you will be learning the language of mathematics in both written and oral form. Once learned, you will be ready to appreciate the beauty of mathematics and to use these skills in all of you future math courses including Abstract Algebra, Topology, Real Analysis, and Complex Analysis.
Motivation:
We will use some interesting,
puzzling, and even amazing examples to motivate you to learn proof
techniques and mathematical theory. In order to prove the statements
it becomes necessary to learn the background material and proper
techniques. Hopefully many of these problems will inspire you to become
more fluent using the tools of mathematics to solve problems. The book
Mathematical Thinking contains many fun examples. We will draw
problems from many mathematical disciplines so that you will also
get an overview of the many different areas of mathematics.
Examples:
The Checkerboard
Problem. Counting squares
of sizes 1 x 1 through 8 x 8, an ordinary 8 x 8 checkerboard has 204 squares.
How can we obtain a formula for the number of squares of all sizes
on an n x n checkerboard?
The Handshake Problem.
Consider n married
couples at a party. Suppose that no person shakes hands with
his or her spouse, and the 2n-1 people other than the host
shake hands with different numbers of people. With how many people
does the hostess shake hands?
Suppose each dot in
an n by n grid of dots is colored black or white.
How large must n be to guarantee the existence of a rectangle
whose corners have the same color?
The Weights Problem.
A balance scale has left and
right pans; we can place objects in each pan and test whether
the total weight is the same on each side. Suppose that five objects
of known integer weight can be selected. How can we choose the weights
to guarantee being able to check all integer weights from 1 through
121?
Nonnegative Integer
Solutions. Suppose that
each resident of New York City has 100 coins in a jar. The coins come
in five types (pennies, nickels, dimes, quarters, half dollars).
We consider two jars of coins to be equivalent if they have the
same number of coins of each type. Is it possible that no two people
have equivalent jars of coin
Chinese Remainder
Problem. A general in ancient
China wanted to count his troops. Suppose that when his soldiers were
split into three equal groups there was one soldier left over, when split
into five equal groups there were two left over, and when split into
seven equal groups there were four left over. Whis is the minimum number
of soldiers that makes this possible?
Calculation
of Grade:
Homework - 25%
Portfolio - 25%
Two 1-hour exams - 25%
Final - 25%
Homework Assignments: (25% of your grade)
I assign many problems
from the book. You are expected to do them all.
From those assigned, I will select some
to hand in. Always write out the entire
question and the solution in complete sentences. These will
be graded and the solutions will be posted. The problems marked (EC)
are optional and will count as extra credit toward your homework grade.
Also, to improve your verbal communication skills,
you will be asked to present the solution of some homework problems
to the class. You should always go over your solution with me during
my office hours before you present it to the class. You will be graded
on your presentation and this will count toward your homework grade.
I will present the remaining problems to
the class or post them on the web.
Try each problem
on your own.
Go over each problem
with someone in the class.
Ask me about the
ones that you still do not completely understand.
Take notes whenever
we go over a problem in class.
Rewrite each problem
as best you can and keep them together in a binder.
Portfolio: (25% of your grade)
From the homework problems
that you do during the semester, I will select 10 for you to rewrite
and hand in as a portfolio of work representing what you learned
in this course. This is intended to be a collection of your best
work. This will be due on Monday December 1st.
You will have two weeks advance notice as to which problems will
be included in the portfolio. The intention here is that you will make
sure you understand how to solve each homework problem as we go along
in case it will be selected to be in the portfolio.
Write out the entire statement of the exercises.
Write the solution as formally as you can. Use complete sentences.
Write it as if it would be published. In fact, I will publish selected
solutions on the web! This should be typed. It is okay to type
the sentences and leave spaces for writing equations by hand. If
you have access to an equation editor, that would be best.