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\usepackage{graphicx}
\usepackage{enumerate}
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\newtheorem{thm}{Theorem}
\newtheorem{ex}{Exercise}

\begin{document}

\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 3\\
\today
\end{flushright}

%Please write your solutions in the designated solution area!

\begin{ex}
Use the the logical equivalences $P\Rightarrow Q \equiv \neg P\vee Q$ and $P\Leftrightarrow Q \equiv (\neg P\vee Q)\wedge(\neg Q\vee P)$ to rewrite
\[
(P\Rightarrow Q)\Leftrightarrow (Q\Rightarrow R).
\]
without $\Rightarrow$ or $\Leftrightarrow$.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Write the {\bf negation} (without just putting the phrase ``It is not the case that...'' in front of the given phrase) and the {\bf contrapositive} of each of the following statements.
\begin{enumerate}[$(a)$]
	\item If today is New Year's Eve, then tomorrow is January.
	\item If the decimal expansion of $r$ is terminating, then $r$ is rational.
	\item If $n$ is divisible by $6$, then $n$ is divisible by $2$ and $n$ is divisible by $3$.
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $P(x): x>3$ and $Q(x):4x-1>12$ be open sentences with domain $D=\{0,2,3,4,6\}$. Determine the truth value of $P(x)\Rightarrow Q(x)$ for each $x\in D$. Give explanations for your answers.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $P(x): x\geq 3$ and $Q(x):4x-1>12$ be open sentences with domain $D=\{0,2,3,4,6\}$. Determine all values of $x\in D$ for which $P(x)\Leftrightarrow Q(x)$ is true.  Give an explanation for your answer.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $P(x): x^2\geq 1$ and $Q(x): x\geq1$ be open sentences with domain $D=\mathbb{R}$. Determine all $x\in D$ for which $P(x)\Rightarrow Q(x)$ is a true statement. Give an explanation for your answer.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $P(x,y): x^2+y^2=1$ and $Q(x,y): x+y=1$ be open sentences where both $x$ and $y$ have domain $\mathbb{Z}$. Determine the truth value of $P(x,y)\Rightarrow Q(x,y)$ for\\ $(x,y)\in\{(1,-1),(-3,4),(0,-1),(1,0)\}$. Give an explanation for your answer. %the \\ there is just so the question looks nicer when compiled. There's nothing special about it.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $D$ denote the set of odd integers and let
\[
P(x): x^2+1 \mbox{ is even.} \quad \mbox{and} \quad Q(x): x^2 \mbox{ is even.}
\]
be open sentences over the domain $D$. State $\forall x\in D, P(x)$ and $\exists x\in D, Q(x)$ in words.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
State the negations of the following quantified statements, without just putting the phrase ``It is not the case that...'' in front of the given phrase.
\begin{enumerate}[$(a)$]
\item For every rational number $r$, the number $\dfrac{1}{r}$ is rational.
\item There exists a rational number $r$ such that $r^2=2$.
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Determine the truth value of each of the following statements. Give explanations for your answers.
\begin{enumerate}[$(a)$]
\item $\exists x\in\mathbb{R}, x^2-x=0$.
\item $\forall n\in\mathbb{N}, n+2\geq 2$.
\item $\forall x\in\mathbb{R}, \sqrt{x^2}=x$.
\item $\exists x\in\mathbb{Q}, 3x^2-27=0$.
\item $\exists x\in\mathbb{R},\exists y\in\mathbb{R}, x+y+3=8$.
\item $\forall x\in\mathbb{R},\exists y\in\mathbb{R}, x+y+3=8$.
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $A$ be the set of circles in the plane with center $(0,0)$ and let $B$ be the set of circles in the plane with center $(1,1)$. Furthermore, let
\[
P(C_1,C_2): \text{$C_1$ and $C_2$ have exactly two points in common.}
\]
be an open sentence where the domain of $C_1$ is $A$ and the domain of $C_2$ is $B$.
\begin{enumerate}[$(a)$]
\item Express the following quantified statement in words:
\begin{equation}
\forall C_1\in A, \exists C_2\in B, P(C_1,C_2). \label{circ}
\end{equation}
\item Express the negation of the quantified statement in \eqref{circ} in symbols.
\item Express the negation of the quantified statement in \eqref{circ} in words.
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\end{document}