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\title{\sc Math 2100 -- Homework 5}
\author{
Fall 2024\\[10pt]
due Wednesday, \textbf{November 13}, \underline{at the beginning of class}\\[10pt]
Sections 2.5, 3.1, 3.2
}
\date{}
\begin{document}
\maketitle
\begin{center}
\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
\end{center}
\textbf{{\Large $\star$} It is not permitted to use any AI tools or Large Language Models (ChatGPT, Claude, Gemini, etc) to assist with this assignment. {\Large $\star$}}
\textbf{Please read the syllabus to remind yourself of our \underline{collaboration policy}.}
\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. It may be handwritten (as long as I can read it) or typed with software such as Word or Latex. Please write the questions in the correct order. Explain all reasoning.
\hrulefill
\underline{You are no longer required to show your scratch work. However, doing scratch work on another paper}\\ \underline{before trying to write your proof is still the way to succeed on these problems!}
\begin{enumerate}
\item Prove that any real number $r$ that makes the equation $r - \ds\frac{1}{r}=5$ true must be irrational.
\item Prove that if $a + b + c \geq 35$, then either $a \geq 10$, $b \geq 12$, or $c \geq 13$.
\item Use Venn Diagrams to determine whether the equation below is true:
\[
(B \cup (A \smallsetminus C)) \cap A \qquad = \qquad A \smallsetminus (A \cap \overline{B} \cap C)
\]
\item Use Venn Diagrams to determine whether the equation below is true:
\[
(\overline{A \cup B}) \cup (\overline{A \cup C}) \qquad = \qquad (B \cup C) \smallsetminus A
\]
\item Write each of the following sets in set-builder notation.
\begin{enumerate}
\item The set $S$ of integers that are multiples of 3 and a perfect square.
\item The set $T$ of positive integers that are bigger than $10$ and whose ones digit is a 5.
\item The set $R$ of real numbers whose square is a rational number.
\end{enumerate}
\item List five elements in each of the following sets, unless there are fewer than 5 elements in the set (in which case, justify how you know you've listed all of the elements).
\begin{enumerate}
\item $A = \{x \in \R : x^2 \in \N\}$
\item $B = \{S \subseteq \{1,2,3,4\} : \text{the sum of the elements of $S$ is even}\}$
\item $C = \{q \in \N : q = 2k \text{ for some $k\in\N$ and }q=2\ell+1\text{ for some $\ell\in\N$}\}$
\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 $A \subseteq B \cap C$.
\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 $B$: $(A \times A) \smallsetminus (B \times B) = (A \smallsetminus B) \times (A \smallsetminus B)$.
\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 any two sets $A$ and $B$: $\PP(A) \cup \PP(B) \subseteq \PP(A \cup B)$.
\end{center}
\end{enumerate}
\end{document}