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%  Math 2001 
%  April 2012
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\begin{center}
{\bf Math 2001}
\end{center}

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Math 2001, Introduction to Discrete Mathematics, 
is a course designed to prepare students for upper division
mathematics classes. Students are expected learn standard mathematical
terminology, foundational material, and methods of proof,
all within the context of discrete mathematics.

Essential topics, which ought to span 1/2---2/3 of a semester, are
\begin{enumerate}
\item[(1)] sets,
\item[(2)] relations,
% and functions.
\item[(3)] logic, and
\item[(4)] proof,
\end{enumerate}
not necessarily in this order.
Secondary topics are often chosen from combinatorics, graph theory,
ordered sets, discrete probability, game theory, 
or discrete geometry. 

\bigskip

\noindent
{\bf Set theory.}
Minimally, students should hear that mathematics is founded
on set theory, which is usually formulated by a short list
of axioms. Students should learn the meanings of the symbols
$\in$, $\emptyset$, $\cup$, $\cap$, ${\mathcal P}(X)$,
as well as the notation $A = \{0,1,2\}$ and $A = \{x\in B\;|\;P(x)\;
\textrm{holds}\}$. Students should learn how to show that 
$A=B$, $A\neq B$, $A\in B$, $A\notin B$, $A\subseteq B$
or $A\not\subseteq B$.

Maximally, it is also useful for students to learn that
each natural number equals the set of its predecessors
and that $\mathbb N$ is defined to be
the smallest set containing
$0$ and closed under successor. 
(This is useful
when discussing induction.)

\bigskip

\noindent
{\bf Relations.}
Minimally, students should learn what an ordered pair is,
what $A\times B$ means, and how relations are defined.
They should learn what a function is, what \emph{injective},
\emph{surjective} and \emph{bijective} mean.
They should learn what equivalence relations are,
and that they are kernels of functions.
They should learn how to determine if
a function is well-defined.
They should learn that mathematical definitions
are used to make concepts definite.

Maximally, students ought to learn that
ordered pairs, cartesian products, relations and functions
are all sets, and that two such objects are equal
if they are equal as sets.
They should learn that composition of functions is associative.
They should learn about the relationship between 
equivalence relations and partitions.
They should learn how to define functions by recursion
(e.g., sum and product of natural numbers).
They should learn what $|A|=|B|$, $|A|\leq |B|$,
$|A|<|B|$, \emph{finite}
and \emph{infinite} mean.

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\noindent
{\bf Logic.}
Minimally, students should hear that 
most of mathematics concerns
declarative statements, which can be combined with 
the conjunctions
\emph{and}, \emph{or}, \emph{not}.
They should learn how a composite statement depends
on its constituents. They should learn about 
quantifiers, and how to use them. 
They should be able to read and write
simple formal statements, and to translate back
and forth between formal and informal statements.
They should be able to negate a formal statement.

Maximally, students should learn about logical equivalence.
It may help to discuss formula trees.
They should learn how to interpret restricted quantifiers,
like $(\forall x P(x))\;(Q(x))$ and $(\exists x P(x))\;(Q(x))$.
(Concrete example: $(\forall x>1) (x^2 > x)$.)
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\noindent
{\bf Proof.}
Minimally, students should hear that a proof is a sequence
of statements in which each statement follows
from preceding statements via accepted rules of deduction.
Students should learn the methods of
direct proof, proof of the contrapositive
and proof by contradiction. Students should be able to prove a simple
statement by any of these methods. They should
learn how a statement differs from its converse.
Students should learn (and practice) induction.

Maximally, students ought to learn why 
direct proof, proof of the contrapositive
and proof by contradiction are equivalent.
Students ought to learn how to determine if
a simple 
statement is true in a structure. (E.g.
is $\forall x\;\exists y\;(x>y)$ true in $\mathbb R$?
What about $\mathbb N$?)
They should learn why induction is a valid form of proof.

\bigskip
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\noindent
{\bf Sample questions.} 
At the end of the course, students ought to be able to 
understand and solve problems like these.
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\begin{enumerate}
\item
Find sets $A$ and $B$ such that $A\in B$ and $A\subseteq B$,
or prove that there are no such sets.
Then find sets $C$ and $D$ such that $C\notin D$ and $C\subseteq D$,
or prove that there are no such sets.
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\item
Prove that if $A\subseteq B$ and $C\subseteq D$, then
$A\cup C\subseteq B\cup D$.
%Prove or disprove: ${\mathcal P}(A\cap B) = 
%{\mathcal P}(A)\cap {\mathcal P}(B)$.
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\item
A theorem with hypothesis $H$ and conclusion $C$ has the form
$H\to C$. Suppose that you can prove that the converse
of the theorem implies the contrapositive of the theorem.
Does this help you to prove the theorem?
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\item
Express the negation of $(\forall x>0)(\exists y(y^2=x))$
in a form where the quantifiers are in front.
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\item
Is $\forall w\;\exists x\;\forall y\;\exists z\;(w^2+x^2=y^2+z^2)$
a true statement about the real numbers? Explain.
\bigskip

\item
Prove or disprove: (a) If $f, g\colon \mathbb R\to \mathbb R$ are both
increasing, then $f\circ g$ is increasing. 
(b)
If $f, g\colon \mathbb R\to \mathbb R$ are both
surjective, then $f\circ g$ is surjective. 
%If $f, g\colon \mathbb R\to \mathbb R$ are both
%decreasing, then $f\circ g$ is decreasing. 
(c)
If $f, g\colon \mathbb R\to \mathbb R$ are both
bounded above, then $f\circ g$ is bounded above.
\bigskip

\item
Let $E$ be an equivalence relation on $X$.
Under what circumstances is
$f\colon X/E\to X\colon [x]_E\mapsto x$ well-defined?
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\item
Show that $\displaystyle 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots
+\frac{1}{\sqrt{n}}<2\sqrt{n}$.
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\item
Show that $(3/2)^n\leq F_{n+2}\leq 2^n$, 
where $F_n$ denotes the $n$th Fibonacci number.
\bigskip

\item
Show by induction that every positive integer is a product of primes.
\end{enumerate}
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