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{\bf Name: }
\hspace{13cm}
{\bf Section: }

\begin{large}
\centerline {\bf Computer Science 243}
\centerline {\bf Spring 2025}
\centerline {\bf Homework 6}
% \centerline{\bf My name}
\vskip .7cm

\bf Due: beginning of class, Wednesday, 3/19/25 \hfill Points: 50
\end{large}
\vskip 0.5cm
Answer the following questions and show your work.  Your final submission must be completely your own work.

\begin{enumerate}

\item {[12 points]}  Let $A = \{x \in {\textbf {\textit R}}\  \vert -1 \le x \le 1\}$ and $B = {\textbf {\textit N}}$.  For each of the following, list the set and state 
if it is finite, countably infinite, or uncountable.
\begin{enumerate}[label=\alph*.]
\itemsep-0mm
\item $A$
\item $A \cap B$
\item $A - B$ 
\item $B - A$ 
\item $B \cup Z^-$
\item $A \cap \overline{A}$
\end{enumerate}

\vskip 0.5cm
\item {[8 points]}  Using the following matrices, compute the matrix operations below:

\begin{align*}
{\bf A} &= 
\begin{bmatrix*}[r]
1 & -2 & 0 \\
3 & 2 & 1 \\
2 & 0 & -1
\end{bmatrix*}
&
{\bf B} = 
\begin{bmatrix*}[r]
2 & 0 & 1 \\
2 & -3 & 0 \\
-1 & 0 & -1
\end{bmatrix*}
\end{align*}

\begin{enumerate}[label=\alph*.]
\itemsep-0mm
\item {[2 points]}  {\bf A + B}
\item {[3 points]}  {\bf AB}
\item {[3 points]}  {\bf BA}
\end{enumerate}

\vskip 0.5cm
\item {[14 points]}  Answer the following using the 0-1 matrices below.

\begin{align*}
{\bf A} &= 
\begin{bmatrix*}[r]
0 & 0 & 1 \\
0 & 1 & 1 \\
1 & 1 & 0
\end{bmatrix*}
&
{\bf B} = 
\begin{bmatrix*}[r]
1 & 1 & 0 \\
1 & 0 & 0 \\
0 & 1 & 1
\end{bmatrix*}
\end{align*}

\begin{enumerate}[label=\alph*.]
\itemsep-0mm
\item $\bf B^t$
\item Is {\bf A} symmetric?  Is {\bf B} symmetric?
\item $\bf A \vee \bf B$ 
\item $\bf A \wedge \bf B$ 
\item $\bf A \odot \bf B$
\item $\bf B \odot \bf A$
\item$\bf A^{[3]}$
\end{enumerate}


\vskip 0.5cm
\item {[16 points]}  Consider the following functions:

\hspace*{1cm} $f_1(n) = 6n^{3/2}$ \\
\hspace*{1cm} $f_2(n) = 64$ \\
\hspace*{1cm} $f_3(n) = 2n^2 + n^3\log n$ \\
\hspace*{1cm} $f_4(n) = 3n \log n$ \\
\hspace*{1cm} $f_5(n) = 6 \sqrt n$ \\
\hspace*{1cm} $f_6(n) = 4n^5 - 7n^4$ \\
\hspace*{1cm} $f_7(n) = n + 2\log n$ \\
\hspace*{1cm} $f_8(n) = 16n^2$

\begin{enumerate}[label=\alph*.]
\item For each function, find the tightest big-O bound.
\item Order the functions from slowest growing to fastest growing (you need only list them by function name, e.g., $f_2$). \\
\end{enumerate}

\end{enumerate}

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