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\centerline{MATH 242 -- Principles of Analysis}
\centerline{Problem Set 1 -- due:  Feb. 1}
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{\it `A` Section}
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\item{1.}  Let $A = \{x \in {\bf R} \mid x^2 - 5x + 6 = 0\}$, $B = (0,4) = \{x \in {\bf R} \mid 0 < x < 4\}$ and $C = \{{x\over x^2 + 1} \mid x \in {\bf R}\}$  (Note:  $C$ is the range of the function $f$ defined by  $f(x) = {x\over x^2 + 1}$.)
\itemitem{a.}  Express the set $C$ as a union of one or more closed intervals $[a,b]$ in ${\bf R}$. (Note:  You should use facts from calculus to solve this.  Don't worry that we have not justified them yet.)
\itemitem{b.}  Find the sets $A \cap C$ and $B\cap C$.
\itemitem{c.}  Find the sets $B\cup A$ and $B\cup C$ and express as unions of intervals in ${\bf R}$.
\item{2.}    Let $B_n = \{1,1/4,1/9,\ldots,1/n^2\}$ for each natural number $n \ge 1$. 
 What are $\cap_{n=1}^\infty B_n$ and
$\cup_{n=1}^\infty B_n$.
\item{3.}  Let $I_n = [-1/n,1/n]$ for any natural number $n \ge 1$.
What are $\cap_{n=1}^\infty I_n$ and $\cup_{n=1}^\infty I_n$?
(Explain your reasoning intuitively.)
\item{4.}  Let $f : {\bf R} \to {\bf R}$ be the function defined by $f(x) = x^2 - 4x + 1$.
\itemitem{a.}  Is $f$ one-to-one?  Why or why not?
\itemitem{b.}  Is $f$ onto?  Why or why not?
\itemitem{c.}  If $I = (1,3)$, what is the set $f(I)$?  Explain.
\itemitem{d.}  If $J = (5,6)$, what is the set $f^{-1}(J)$.  Explain.
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{\it `B' Section}
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\item{1.}  Prove part (f) of Theorem 1.1.3 in the text.  These are the {\it De Morgan Laws} for
complements.
\item{2.}  Let $A$ and $B$ be arbitrary sets.  Does $B = A - (A - B)$, as we might expect
if we looked at the formula through the lens of ordinary algebra?  If this is always true,
prove it; if it is not, give both a counterexample (an example where the formula is not true),
and a correct statement with proof.
\item{3.}  Let $f : A \to B$ be a function.
\itemitem{a.} Let $C, D$ be subsets of $A$.  Is it always
true that $f(C\cup D) = f(C) \cup f(D)$?  If this is always true prove it; if it is not,
give a counterexample.
\itemitem{b.}  Show that $f$ is onto if and only if $f(f^{-1}(E)) = E$ for all subsets $E$ of $B$.
\item{4.}  Let $f : A \to B$ and $g : B \to C$.
\itemitem{a.}  Show that if $f$ and $g$ are both one-to-one, then $g \circ f : A \to C$ is
also one-to-one.
\itemitem{b.}  Is the converse of the statement in part a true?  That is, if you know
that $g \circ f$ is one-to-one, does it follow that $f$ and $g$ are one-to-one?  Prove
or find a counterexample.
\bye 