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{{\Large\bf MODERN ALGEBRA I}\\
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{\Large\bf HOMEWORK ASSIGNMENT I}\\
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{\bf (Turn in underlined problems.)}}
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{\bf 
\begin{tabbing}
\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

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\noindent
1 \> 1.5, \underline{1.9}, \underline{1.16} (Give the number.),
\underline{1.18}
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\underline{\large\bf ADDITIONAL PROBLEMS}
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{\bf 

\underline{1.} Show by induction that 
$\displaystyle 2^n \leq
\left(\begin{array}{c}2n\\n\end{array}\right)
\leq 4^n$ for $n\geq 0$.

\medskip

2. Recall from linear algebra how to compute the 
determinant of a matrix using cofactor expansion.
Using this method for computing the determinant, prove 
(by induction) that 
the determinant of the $n\times n$ identity matrix is $1$.

\medskip

\underline{3.} For real numbers $r\geq 0$ let $S_r$
be the statement \[S_r:\;r=0.\]
Show that
\begin{enumerate}
\item[$(a)$] $S_0$ holds.
\item[$(b)$] If $S_r$ holds for all $r < r_0$, then $S_{r_0}$ holds.
\item[$(c)$] Some of the $S_r$'s do not hold.
\end{enumerate}
What conclusion about induction can be drawn from
these observations?

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