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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Exam 1}
\centerline{February 15, 2013}
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\noindent
{\it General Information}
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The first exam for the course will be given in class on
Friday, February 22.  This will be an individual, in-class, closed 
book exam.  I will be happy to hold a late afternoon or evening review 
session to help you prepare.  
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\noindent
{\it What to Expect}
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The exam will have four or five questions, each possibly with several
parts.   Some questions will ask for a precise statement of a 
definition or a theorem we have discussed.  I might ask you
to give one of the proofs of key results we have covered.  
The Sample Questions below should give you 
some idea what they might look like.  (The actual exam questions
will be different of course!)
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\noindent
{\it Topics to be Covered}
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The exam will cover the material we have covered since the start of the 
semester up to and including class on Friday, February 15.  This includes
all of the material from Chapter 1 and section 2.1 in the text:
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\item{1)}  Sets, set operations (unions, intersections, complements)
\item{2)}  Functions, the one-to-one and onto properties  
\item{3)}  The Well Ordering Property of ${\bf N}$ and the 
technique of proof by mathematical induction.  Application to the binomial theorem.
\item{4)}  Algebraic and order properties of ${\bf R}$. Know the proof that $\sqrt{2} \notin {\bf Q}$.
\item{5)}  Upper and lower bounds, least upper bound and
greatest lower bound.  The Least Upper Bound Axiom for the real 
number system: Know the statement and what it means. 
\item{6)}  Consequences of LUB Axiom (know these proofs):
\itemitem{(a)}  ${\bf N}$ is not bounded in ${\bf R}$.
\itemitem{(b)}  If $\varepsilon > 0$, then there exists $n \in {\bf N}$ such
that $0 < {1\over n} < \varepsilon$.  
\itemitem{(c)}  If $a < b$ in ${\bf R}$, then 
there is a rational number $r \in {\bf Q}$ such that
$a < r < b$ (and hence infinitely many) and also an irrational $s \in {\bf Q}^c$ such
that $a < s < b$ (and hence infinitely many).   
\item{5)}  Sequences and the definition of convergence. 
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\noindent
{\it Practice Questions}
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\noindent
{\it Don't be too concerned about the length of this list.  
The actual exam will be somewhat less than half the total length of these questions.
The idea is to show the range of different types of questions and
topics that might be covered.  }
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\noindent
I.  Let $A = [1, 5)$, $B = (-1,2]$, and $C = (-1,1)$.
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\item{A)}  What are $A \cup B$, $A \cap B$, $A - B = A \cap B^c$?
\item{B)}  What are ${\rm lub}(A)$, ${\rm lub}(B)$?
\item{C)}  Let $C = \{x\cdot y \vert x \in A, y\in B\}$.
Identify ${\rm lub}(C)$ and prove your claim.
\item{D)}  Let $D = \{1/x \vert x \in A\}$.
What are ${\rm lub}(D)$ and ${\rm glb}(D)$?  Give an explanation
how you know your claims are true.   
\item{E)}  Prove that if $E$ is any bounded set of strictly
positive real numbers, and $F = \{1/x \vert x \in E\}$,
then ${\rm lub}(F) = 1/{\rm glb}(E)$.
\item{F)}  Is there a one-to-one and onto function
$f: {\bf R} \to C$?  If so find one; if not say why not.
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\noindent
II.  
\item{A)}  Prove by induction:  $2n^3 + 3n^2 + n$ is evenly divisible by $6$ for all $n \ge 1$.  
\item{B)}  Prove by induction:  ${3^n \over n!} > 1$ for all natural numbers $n \ge 7$.  
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\noindent
III.  Saying a function $f : {\bf R} \to {\bf R}$
is strictly increasing means: for all real numbers
$x$ and $y$ satisfying $x < y$, the inequality $f(x) < f(y)$ holds.  
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\item{A)} Give the {\it negation} of the statement 
$f$ is strictly increasing in the same sort of form as the definition
above.  (What does it mean for that statement to be false?
Just saying ``$f$ is not strictly increasing''
is not an acceptable answer!)
\item{B)}  Give the contrapositive of the statement: ``If $f$ is 
one-to-one, then $f$ is strictly increasing.''  Is that a true statement?
\item{C)}  Consider $f : {\bf R} \to {\bf R}$
defined by $f(x) = x^2$.  What is wrong with the following
reasoning?  If $x < y$, then $x^2 < y^2$; therefore
$f(x) = x^2$ is a strictly increasing function.
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\noindent
IV.  
\item{A)}  State the Least Upper Bound Axiom for the real number system.
\item{B)}  Show that for any real number
number $y > 1$, there exists a natural number $n$ such 
that $1 + {1\over n} < y$.
\item{C)}  Let $n$ represent a natural number and let
$A = \cap_{n=1}^\infty \left[1,1+{1\over n}\right]$.  
Is $A$ the empty set?  Why or why not?
\item{D)}  Let $B = \cup_{n=1}^\infty \left(0,1+{1\over n}\right)$.
What is ${\rm lub}(B)$?  Explain.
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V.  
\item{A)}  Prove that if $a < b$ are any real numbers, then there is some
$r \in {\bf Q}$ such that $a < r < b$.
\item{B)}  Show that if $r$ is a rational number, then
$\sqrt{2} + r$ is an irrational number.  (Argue by contradiction.)
\item{C)}  Show that if $a < b$ are any real numbers, then there
is an irrational number $s$ with $a < s < b$. 
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\noindent
VI.  
\item{A)}  Show using the definition that the sequence $x_n = {n + 1\over 3n + 2}$
converges to ${1\over 3}$. 
\item{B)}  Show using the definition that the sequence $x_n = {1 + (-1)^n}$ does 
not converge.   

\bye
