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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Midterm Exam 1 -- October 1, 2004}
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{\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.
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I. (10) Construct the truth table for the compound expression
$$ (p \Rightarrow q) \Leftrightarrow (\sim p \vee q).$$
What does your result say?
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II.  Consider the following definition:  A function $f$
is periodic if and only if there exists a real number
$P > 0$ such that for all real numbers $x$, $f(x + P) = f(x)$.
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\item{A)} (5) Consider $f : {\bf R} \to {\bf R}$
defined by $f(x) = x^2$.  If $P = 2$, then notice 
$f(-1 + 2) = f(-1)$ is true.  Does this say $f(x) = x^2$ is
periodic?  Why or why not?
\item{B)} (10) Give the negation of the statement 
``$f$ is periodic''.
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III.  Let $A = \{x \in {\bf R} : 1 < x^2 < 9\}$, let $B$ be the
interval $[-2,4)$, and let $C = \{1,2,3\}$.
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\item{A)}  (10) What is the set $A\cap B$?
\item{B)}  (10) What is the set $(A\cup B)\ \backslash\ C$?
\item{C)}  (10) Let $f: {\bf R} \to {\bf R}$ be defined
by $f(x) = -x + 3$.  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.
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IV. (20) Show that the open interval $(0,1)$ in the real numbers
is not a denumerable set.
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V.  (15) Prove that for all $n \in {\bf N}$, 
$$1^3 + 2^3 + \cdots + n^3 = {1\over 4}n^2(n+1)^2.$$
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{\it Extra Credit}
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(10)  Let $x$ be a non-negative real number.  Show that if
$x \le \varepsilon$ for all $\varepsilon > 0$, then $x = 0$.
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