\magnification=\magstep1
\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Makeup Midterm Exam 1 -- October 7, 2004}
\bigskip
\bigskip
\noindent
{\it Directions:}  Do all work in the blue exam booklet.  There are
100 possible regular points (distributed as indicated in the questions
below), and 10 Extra Credit points.
\bigskip
\noindent
I. (10) Construct the truth table for the compound expression
$$( (\sim q) \wedge (p \Rightarrow q)) \Rightarrow\ \sim p.$$
What does your result say?
\bigskip
\bigskip
\noindent
II.  Consider the following definition:  A function $f$
is strictly increasing if and only if for all real $x$ and  
all real $y$, 
if $x < y$ then $f(x) < f(y)$.  
\bigskip
\item{A)} (5) 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.
\item{B)} (10) Give the negation of the statement 
``$f$ is strictly increasing''.
\bigskip
\bigskip
\noindent
III.  Let $A = \{x \in {\bf R} : 0 < x^2 - 1 < 3\}$, let $B$ be the
interval $[-1,2)$, and let $C = \{-1,2\}$.
\bigskip
\item{A)}  (10) What is the set $A\cup B$?
\item{B)}  (10) What is the set $(A\cap B)\ \backslash\ C$?
\item{C)}  (10) Let $f: {\bf R} \to {\bf R}$ be defined
by $f(x) = 2x + 1$.  What is the set $f^{-1}(A)$?
\item{D)}  (10) Is the function $f$ from part C injective?  Is it 
surjective (onto ${\bf R}$)?  Justify your answers.
\bigskip
\bigskip
\noindent
IV. (20)  Show that the set of positive and negative integers,
${\bf Z}$, is denumerable.
\bigskip
\bigskip
\noindent
V.  (15) Prove that for all $n \in {\bf N}$, 
$$1 + 3 + 5 + \cdots + (2n - 1) = n^2.$$
\bigskip
\noindent
{\it Extra Credit}
\bigskip
\noindent
(10) Show that if $f : A \to B$ and $g : B \to C$ are
injective mappings, then $g \circ f : A \to C$ is also 
injective.
\bye
