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\centerline{MATH 242 -- Principles of Analysis}
\centerline{Problem Set 1 -- due:  Jan. 31}
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{\it `A` Section}
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\item{1.} Assume that $A,B$ are sets of integers.
\itemitem{a.}  What is the contrapositive of the statement: ``If $x$ is even then $x \in A \cup B$''?  Express without
using {\it not}.
\itemitem{b.}  What is the converse of the statement in part a?  
\item{2.}  Let $A = \{x \in {\bf R} \mid x^2 - 5x + 4 = 0\}$, $B = (0,1) = \{x \in {\bf R} \mid 0 < x < 1\}$ and $C = \{{x\over x^2 + 9} \mid x \in {\bf R}\}$  (Note:  $C$ is the range of the function $f$ defined by  $f(x) = {x\over x^2 + 9}$.)
\itemitem{a.}  Express the set $C$ as a closed interval $[a,b]$ in ${\bf R}$ or as a union of such intervals. (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 $B \cap A$ and $B\cap C$.
\itemitem{c.}  Find the sets $A \cup B$ and $A \cup C$ and express using set notation.
\item{3.}    For $n$ a general natural number, let $B_n = \{0,2n\}$.
 What are $\cap_{n=1}^\infty B_n$ and $\cup_{n=1}^\infty B_n$?
\item{4.}  Let $I_n = [-1/n,1/n]$ for a general 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{5.}  Let $f : {\bf R} \to {\bf R}$ be the function defined by $f(x) = \tan^{-1}(x)$.
\itemitem{a.}  Is $f$ one-to-one?  Why or why not?
\itemitem{b.}  Is $f$ onto?  Why or why not?
\itemitem{c.}  If $I = (0,\sqrt{3})$, what is the set $f(I)$?  Explain.
\itemitem{d.}  If $J = (-\pi/4,\pi/4)$, 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 DeMorgan Laws} for
unions and intersections.
\item{2.}  Let $A$ and $B$ be arbitrary sets.  Does $A = A - (B - B)$, 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).
\item{3.}  Let $f : A \to B$ be a function.
\itemitem{a.} Let $C, D$ be subsets of $A$.  Is it always
true that $f(C\cap D) = f(C) \cap f(D)$?  If this is always true prove it; if it is not,
give a counterexample.
\itemitem{b.}  Show that $f$ is one-to-one if and only if $f^{-1}(f(C)) = C$ for all subsets $C$ of $A$.
\item{4.}  Let $f : A \to B$ and $g : B \to C$.
\itemitem{a.}  Show that if $f$ and $g$ are both onto, then $g \circ f : A \to C$ is
also onto.
\itemitem{b.}  Is the converse of the statement in part a true?  That is, if you know
that $g \circ f$ is onto, does it follow that $f$ and $g$ are onto?  Prove
or find a counterexample.
\bye 