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\title{\sc Math 2100 -- Homework 2}
\author{
	Fall 2024\\[10pt]
	due Wednesday, \textbf{October 2}, \underline{at the beginning of class}\\[10pt]
	Sections 1.4, 1.5
}
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\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. There are tutorials and templates for using Latex on our course website. If you type your homework, you can include hand drawn pictures in it. To do this in Latex, use the ``$\backslash$includegraphics'' command (see how I do it in this document). Please write the questions in the correct order. 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:
\begin{enumerate}[label={$\diamond$}]
	\item explain your reasoning
	\item don't leave out steps
	\item 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 $3 \not\in \overline{S}$''.
\end{enumerate}

\hrulefill

\begin{enumerate}
	
	\item Write each of the sentences below using predicates and quantifiers. Each one tells you how many quantifiers your answer should have. Make sure you define each set clearly (like we did in class).
	\begin{enumerate}
		\item Every exam has at least one typo. --- 2 quantifiers
		\item Some cats do not like having their belly rubbed. --- 1 quantifier
		\item Nobody likes prunes. --- 1 quantifier
		\item Some books have characters that nobody likes. --- 3 quantifiers
	\end{enumerate}
	
	\item For each of the four sentences above, write the negation. (Note: It will be very helpful to start with your previous answer that you put in predicate-quantifier form!)
	
	\item Write the negation of each of these sentences.
	\begin{enumerate}
		\item Every time you roll a ``6'', you have to take a card.
		\item There is a Marquette student who gets A's in all of her classes.
		\item In every good book, there is a plot twist or surprise ending.
		\item Every math course has a topic that everyone finds easy to do.
	\end{enumerate}

	
		
	\item Give your own example, different from the ones in class, of a predicate $P(x,y)$ such that $\forall x, \exists y, P(x,y)$ and $\exists y, \forall x, P(x,y)$ mean different things. Explain what each version means. (Each student in the class should have a different answer.)
		
	\item For each statement below, translate from English to Math, using predicates, quantifiers, and implications where necessary. \emph{The actual statements might be true or false, so you don't need to prove any of these!}
	\begin{enumerate}
		\item For all real numbers $r$, if $r^2 > 0$ then $r > 0$.
		\item There exists a real number $m$, such that for all rational numbers $q$, $m = q^2$.
		\item For all natural numbers $n$ and $k$, if $n$ is a multiple of $k$, then $n^2$ is a multiple of $k^2$.
	\end{enumerate}
	
	\item Write the negation of each implication in English. It may help to translate to math first.
	\begin{enumerate}
		\item If it is snowing, then it is cold outside.
		\item If it is cold outside, then it is snowing.
		\item If I have three cats and a dog, then I have too many pets!
		\item If I go to sleep too late or I eat ice cream for breakfast, then I don't feel good.
	\end{enumerate}
	
	\item Express each of the following statements using predicates and quantifiers. (You do not need to prove them!)
	\begin{enumerate}
		\item If $n$ is a multiple of $5$, then $n$ ends in $5$ or $n$ ends in $0$.
		\item If $n$ is not a multiple of $3$, then $n^2 - 1$ is a multiple of $3$
		\item For all odd integers $a$ and $b$, there is no real number $x$ such that $x^2 + ax + b = 0$.
		\item For every real number $y$, if $y \geq 0$, then there exists $x \in \R$ such that $x^2 = y$.
	\end{enumerate}
	
	\item For parts (a) and (b) in the previous question, write the converse, inverse, and contrapositive. (Be sure to label which is which.)

	
	\item Use a truth table to determine if the two propositions are logically equivalent:
	\[
		q \to ((p \to r) \wedge (r \to p)) \hspace*{1in} q \wedge r
	\]
	
	\item For each part below, devise an implication that satisfies the conditions of that part (not all parts at once), or if it's not possible, explain why not. Your answer should be different from any of the examples we did in class.
	\begin{enumerate}
		\item an implication that is false, but its converse is true
		\item an implication that is false, but its contrapositive is true
		\item an implication that is false, and its converse is false
	\end{enumerate}


\end{enumerate}
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