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\title{\sc Math 2100 / 2350 -- Homework 2}
\author{
Fall 2019\\[10pt]
due Wednesday, \textbf{October 2}, at the beginning of class\\[10pt]
Sections 1.4, 1.5, 3.1, 3.2
}
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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. \textbf{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. Explain all reasoning.
\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 $3 \not\in \overline{S}$''.
\begin{enumerate}
\item Negate the following statements.
\begin{enumerate}
\item If it is raining, then there is lightning.
\item If $n$ is a natural number, then either $n$ is even or $n$ is odd.
\end{enumerate}
\item Use a truth table to determine whether the two statements are equivalent.
\[
(p \vee q) \to (p \wedge q) \hspace{1in} p \to q
\]
\item Form a predicate and a quantified statement that represents the following sentence: ``Every university has a dorm that is at least 20 years old.''
\item Negate the statement ``There is a car that everyone wants to buy.''
\item Use Venn Diagrams to determine whether the equation below is true:
\[
(A \cup B) \smallsetminus (A \cap C) = B \cup (A \smallsetminus C).
\]
\item List 5 elements of each of the following sets, unless there are fewer than $5$ elements (in which case, list them all and justify how you know you've listed all of them).
\begin{enumerate}
\item $\{x \in \R : x \not\in \N \text{ and } x^2 \in \N\}$
\item $\{ S \subseteq \N : \text{the sum of the elements in $S$ is less than $3$}\}$
\item $\{ z \in \N : z = 5k+2 \text{ for some }k \in \Z\}$
\item $\{ r \in \Z : r = 2k \text{ for some $k \in \Z$ and } r = 2\ell+1 \text{ for some $\ell \in \Z$}\}$
\end{enumerate}
\item Write each of the following sets in set-builder notation.
\begin{enumerate}
\item The set $A$ of real numbers that are not rational numbers.
\item The set $B$ of rational numbers whose numerator is $1$ and whose denominator is a prime number.
\item The set $C$ of pairs of real numbers $(r_1, r_2)$ that add up to a natural number.
\item The set $D$ of subsets of the real numbers whose size is $10$ or less.
\end{enumerate}
\item Determine whether the statement below is true or false. If true, give a few sentences of justification (a formal proof is not necessary). If false, give specific examples of sets that make the statement false.
\begin{center}
For all sets $A$, $B$, and $C$: if $A \subseteq B$ and $A \subseteq C$, then $A \subseteq B \cup C$.
\end{center}
\item Determine whether the statement below is true or false. If true, give a few sentences of justification (a formal proof is not necessary). If false, give specific examples of sets that make the statement false.
\begin{center}
For all sets $A$ and $B$: $(A \cup B) \times (A \smallsetminus B) = (A \times A) \smallsetminus (B \times B)$.
\end{center}
\item Determine whether the statement below is true or false. If true, give a few sentences of justification (a formal proof is not necessary). If false, give specific examples of sets that make the statement false.
\begin{center}
For all sets $A$ and $B$: if $A \subseteq B$ then $A \subseteq \PP(B)$.
\end{center}
\end{enumerate}
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