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\title{\sc Math 2100 / 2105 / 2350 -- Homework 2}
\author{due Thursday, September 13, at the beginning of class}
\date{}
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\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
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\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Staple your work together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive credit.
\textbf{Mathematical Writing:} An important component of this course is learning how to write mathematics correctly and concisely. Your goal should always be the convince the reader that you are correct! That means explaining your thinking and each step in your solution. We will talk more about this when we cover formal proofs in a few weeks, but for now I expect you to do the following: explain your reasoning, don't leave out steps, and use full sentences with correct spelling and grammar (including your use of math symbols). For example, don't write ``$3 \in S \Longrightarrow 3 \not\in \overline{S}$''; instead, write ``Since $3 \in S$, it follows that $3 \not\in \overline{S}$''.
%\emph{You may use Wolfram Alpha or another similar tool to compute any necessary sums or integrals, and for your matrix calculations. If you have trouble with this, let me know.}\\
%\textbf{If you're using facts about distributions to answer the questions, be very clear about which distribution you're using to model that problem and why that distribution is appropriate.}
%\underline{\textbf{Every step of your answers must be fully justified to receive credit.}}
\begin{enumerate}
\item Use a Venn diagram to determine whether each of the following set equality is true or false. If true, explain why (use your Venn diagram---I'm not asking for a formal proof yet); if false, give examples of sets for which the two sides are not equal.
\[
A \cap (B \cup C) = (A \cap B) \cup C
\]
\item Determine whether the statement below is true or false. If true, give a few sentences of justification (not a formal proof). If false, give specific examples of sets for which the two sides are not equal.
\begin{center}
For all sets $A$, $B$, and $C$: if $B \subseteq C$, then $A \times B \subseteq A \times C$.
\end{center}
\item Determine whether the statement below is true or false. If true, give a few sentences of justification (not a formal proof). If false, give specific examples of sets for which the two sides are not equal.
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For all sets $A$, $B$, and $C$: $(A \cup B) \times (A \smallsetminus B) = A^2 \smallsetminus B^2$.
\end{center}
\item In New Hampshire, license plates contain exactly $8$ letters with all of the following properties:
\begin{enumerate}[i)]
\item The first and third letters are vowels.
\item All other letters are consonants.
\item No letter is repeated more than once.
\item The letter ``Y'' is considered a consonant, not a vowel.
\item The license plate does end in ``X''.
\end{enumerate}
How many different license plates can be made that obey these rules?
\item You roll a six-sided die five times. How many possible outcome sequences are there in which you do not roll the same number twice in a row? For example, $14146$ and $64141$ both count, but $11464$ does not.
\item A six-digit number is a number between $100000$ and $999999$. How many six digit numbers are there in which either all of the digits are even, or all of the digits are odd? For example, $517931$ counts, but $214365$ does not, and $022446$ is not even a valid six-digit number.
\end{enumerate}
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