\magnification=\magstephalf
{\bf \centerline{Math 223 Final Exam;  May 3, 2004}}\medskip
\noindent{\bf Instructions for questions 1--8.} Answer the questions 
in the spaces provided. They are worth {\sl 5 points each}.
\medskip

\noindent{\bf 1.}  Consider the permutations of the set $\{1,2,3,4\}$
  given in two-line notation by
  $$g=\biggl({1\atop 3}{2\atop
4} {3\atop 1} {4\atop 2}\biggr),\qquad h=\biggl({1\atop 3} {2\atop 2} 
{3\atop 1}
{4\atop 4}\biggr)$$  (recall this means $g(1)=3,g(2)=4,\ldots$ etc).
  Calculate the composite function $g\circ h$, expressing your answer 
in two-line notation.\vskip 3.5cm




\noindent{\bf 2.} Find all integer solutions  $x$, $y$ of the linear 
Diophantine equation
$30x +24y=18$.\vskip 3.2cm

\noindent{\bf 3.} Use limit laws to determine the limit of the 
sequence $\langle{{2\cdot 4^n+2^n}\over{3^n+3\cdot4^n}}\rangle$
  as $n\rightarrow \infty$.\vskip 3.2cm

\noindent{\bf 4.}  Determine the least upper bound of the subset $A$ 
of $\bf R$ defined by
$$A:=\{\,x\in {\bf{R}}\mid -1\leq x\leq 2\}\cup\{3-{200\over n}\mid 
n\in {\bf N}\}.$$ \vskip 3.2cm
\vfill\eject
\noindent{\bf 5.} Find all the distinct solutions (modulo 10) of the equation
$x^2-x-2\equiv 0\pmod{10}.$\vskip 4cm
\noindent{\bf 6.} The function $f\colon {\bf R}\rightarrow {\bf R}$ 
is defined by $f(x)=x^3-2$.
Determine whether $f$ is a bijection and if so, give an explicit 
formula for the inverse function $f^{-1}$.
\vskip 4 cm
\noindent{\bf 7.} Calculate the inverse $[25]^{-1}$ of the congruence 
class of $25$ in ${\bf Z}_{1001}$ (the integers modulo $1001$).
\vskip 4cm
\noindent{\bf 8.}  Determine the integer $a$ with $0\leq a\leq22$ and 
$2^{100}\equiv a\pmod {23}$. \vskip 4cm
\vfill\eject
{\bf Instructions for Questions 9--14} Give careful statements of the 
indicated definitions, principles or theorems in the spaces provided.
These questions are worth {\sl 5 points each}. Points will be 
deducted for unclear or imprecise
  (or of course incorrect) answers.\bigskip

\noindent{\bf 9.} Let $\sim$ be an equivalence relation on a set $X$.
Give the definition of the equivalence class $[x]$ of an element $x$ of $X$.
  If $a,b,c\in X$ with
$c\in [a]\cap [b]$, what can you conclude about $[a]$ and $[b]$?
\vskip 5 cm
\noindent{\bf 10.} Explain what is meant by saying
  that a sequence $\langle{a_n}\rangle$ of real numbers converges to a 
limit $L$.
\vskip 5cm
\noindent{\bf 11.} Explain what is meant by a lower bound, and the 
greatest lower bound, of  a subset $A$ of $\bf R$.
Carefully state the greatest lower bound property of $\bf R$. \vskip 
5cm\vfill\eject
\noindent{\bf 12.} Give a careful definition of the notion of a 
Cauchy sequence.\vskip 6cm

\noindent{\bf 13.} State the Archimedean property of $\bf R$.\vskip 6cm
\noindent{\bf 14.} State the Bolzano-Weierstrass theorem.

\vfill\eject
{\bf Instruction for Questions 15--18} Questions 15--18 are 
True/False questions worth {\sl 5 points each}.
Write your answer (T or F) in the space provided. No working is 
required.\bigskip
\noindent{\bf 15.} If $f\colon A\rightarrow B$ and $g\colon 
B\rightarrow C$ are functions such that
$g\circ f$ is injective, then $f$ is injective.
\bigskip Answer:\vskip 3cm
\noindent{\bf 16.} Any subsequence $\langle b_n\rangle$  of a Cauchy 
sequence $\langle a_n\rangle$
is convergent.\bigskip Answer: \vskip 3cm
\noindent{\bf 17.} If $N\geq 2$ is any integer and $a\in{\bf Z}$ is 
not divisible by $N$,
then $a^{N-1}\equiv 1\pmod N$. \bigskip Answer:
\vskip 3cm
\noindent{\bf 18.} The congrunce
$ax\equiv 1\pmod p$ has a unique solution (modulo $p$) for $x$ if $p$ 
is a prime number
  and $a$ is an integer not divisible by $p$.\bigskip Answer: \vskip 3cm


\vfill\eject
{\bf Instruction for Questions 19--24} Provide careful proofs  of the
  indicated statements in the spaces provided. Points will be deducted 
for unclear or
  imprecise explanation. Each question is worth {\sl 10 points}.\medskip

\noindent{\bf 19.} Prove by induction on $n$  that $1+2+3+\cdots 
+n={1\over 2}n(n+1)$
for $n\in {\bf N}$.\vskip 5cm
\noindent{\bf 20.} Prove that if the sequence $\langle a_n\rangle $ 
is convergent,
it is a Cauchy sequence.\vskip 6cm
\noindent{\bf 21.}  Prove that $\sqrt 3$ is irrational.\vskip 5cm\vfill\eject
\noindent{\bf 22.}  Prove that any convergent sequence is bounded.\vskip 6cm
\noindent{\bf 23.} State and prove the rational zeros theorem, on the 
rational zeros
  of a polynomial with integer coefficients.\vskip 7cm
\noindent{\bf 24.}  Prove that if $a$, $L$ are real numbers such that 
$a\geq L-\epsilon$
  for all real numbers $\epsilon >0$, then
$a\geq L$.\vskip 5cm

\vfill\eject\bye






\noindent{\bf 2.} Determine the least natural number $N$ such that 
$2^N\equiv 1\pmod {25}$.
\vskip 4cm
\noindent{\bf 3.}  Find a solution of the three simultaneous congruences
$x\equiv 1\pmod 2$, $x\equiv 3\pmod 5$ and $x\equiv 2\pmod 7$. \vskip 4cm

\noindent{\bf 4.} Determine all the rational zeros of the polynomial 
$f(x)=x^3-3x^2+4$.

\vfill\eject
{\bf Instructions for Questions 5--8} Give careful statements of the 
indicated definitions, principles or theorems in the spaces provided.
These questions are worth {\sl 5 points each}. Points will be 
deducted for unclear or imprecise
  (or of course incorrect) answers.\bigskip
\noindent{\bf 5.} Give a careful definition of a prime natural 
number, and state the fundamental
  theorem of arithmetic.
\vskip 3.8 cm
\noindent{\bf 6.} Let $n\geq 2$ be a natural number. Explain what is 
meant by a congruence class modulo $n$. How many such (distinct) 
congruence classes are there?
\vskip 3.8cm
\noindent{\bf 7.}  State Fermat's little theorem.\vskip 4cm


\noindent{\bf 8.} Recall the construction of the rational numbers 
from the integers
  (involving  the set $F$ of ordered pairs of integers $(a,b)$ with 
$b\neq 0$ under
  a suitable equivalence relation $\sim$).
Describe the appropriate equivalence relation $\sim$ on $F$ 
explicitly. How is the fraction $a/b$ defined explicitly
  in terms of $F$ and $\sim$?\vskip 3cm

\vfill\eject
{\bf Instruction for Questions 9--12} Questions 9--12 are True/False 
questions worth {\sl 5 points each}.
Write your answer (T or F) in the space provided. No working is 
required.\bigskip
\noindent{\bf 9.} The only solutions $x\in \bf Z$  of the equation 
$x^2\equiv 1\pmod 8$ are $x\equiv 1\pmod 8$ and $x\equiv -1\pmod 8$.
\bigskip Answer:\vskip 3cm
\noindent{\bf 10.} Let $X=[8]=\{1,2,3,\ldots,8\}$. Consider the 
partition of $X$ into the sets
$\{1,3,5,7\}$,  $\{2,4,6,8\}$. Then the   equivalence relation $R$ on 
$X$ corresponding to this partition is given
by $xRy$ iff $x,y\in X$ and $x-y$ is even.  \bigskip Answer: \vskip 3cm
\noindent{\bf 11.}  The congruence $57 x\equiv 1 \pmod {403}$ has a 
unique solution.
\bigskip Answer: \vskip 3cm
\noindent{\bf 12.} The set ${\bf Z}_n$ of integers  modulo $n$ forms 
a group with binary operation $\circ$ given by
  $[a]\circ [b]=[ab]$. \bigskip Answer:
\vskip 3cm

\vfill\eject
{\bf Instruction for Questions 13--15} Provide careful proofs  of the
  indicated statements in the spaces provided. Points will be deducted 
for unclear or
  imprecise explanation.\bigskip
\noindent{\bf 13.} {\sl (15 points)} Prove that there are infinitely 
many prime numbers.\vskip 6.5cm
\noindent{\bf 14.} {\sl (10 points)} Prove: if $a$, $b$, $c$ are integers such
  that $a$ and $b$ are relatively prime  and $a$ divides $bc$, then 
$a$ divides $c$ \vskip 6.5cm
\noindent{\bf  15.} {\sl (15 points)} Prove that the relation defined 
on ${\bf R}\times {\bf R}$ by $(x,y)\sim
(a,b)$ if $x^2+y^2=a^2+b^2$ is an equivalence relation.
\vfill\eject\bye




\noindent{\bf 1.} According to the division algorithm for $\bf Z$, 
what is the quotient and remainder when
$55$ is divided by $-8$?\vskip 2cm

\noindent{\bf 2.} Consider the expansion 
$$(x+y)^8=ax^8+bx^7y+cx^6y^2+\cdots$$ given by the binomial theorem.
Express $a,b,c$ as binomial coefficients and determine their explicit 
numerical values
  (You do NOT have to evaluate the full expansion, just the part above 
involving $a$, $b$ and $c$).
\vskip 4cm

\noindent{\bf 4.} If functions $f,g\colon \bf R\rightarrow \bf R$ are 
defined by $f(x)=x-1$ and $g(x)=x^2-1$ for $x\in \bf R$,
compute $(g\circ f)(x)$ for $x\in \bf R$.
\vfill\eject
