\documentclass[12pt,reqno]{amsart}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
\usepackage{circuitikz}
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\newtheorem{thm}{Theorem}
\newtheorem{ex}{Exercise}
\begin{document}
\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 2\\
\today
\end{flushright}
\begin{ex}
Determine which of the following are statements. If not, explain why not. If so, determine the truth value of the statement.
\begin{enumerate}[$(a)$]
\item Calvin Coolidge was the greatest American President.
\item The square root of a rational number is always a rational number.
\item $1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\dfrac{n(n+1)}{2}\right)^2$.
\item Mixing yellow and red paint will give you orange paint.
\item Life is like a box of chocolates.
\item When will the Red Sox win the World Series?
\item This sentence is false.
\item A group of owls is called a parliament.
\item $\dfrac{n(n+1)}{2}$.
\item Every former President of the United States is buried in the United States.
\item Everyone has a cat.
\end{enumerate}
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Write the negation of each of the following statements, without just putting the phrase ``It is not the case that...'' in front of the given phrase.
\begin{enumerate}[$(a)$]
\item Pi is a positive real number.
\item Georgia is the eleventh largest state.
\item Flatland State University has no major in paleontology.
\item $3+1<4$.
\item $3$ is a factor of $7$.
\item $1+2+3 = \dfrac{3(3+1)}{2}$.
\item Sam is an orange belt and Kate is a red belt.
\item The train is late or my watch is fast.
\end{enumerate}
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Write a truth table for the statement
\[
\neg p \wedge q.
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Write a truth table for the statement
\[
p \wedge (\neg q \vee r)
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Determine whether the given pair is logically equivalent. Justify your answer using truth tables and include a few words of explanation.
\[
p\vee (p\wedge q) \text{ and } p.
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Determine whether the given pair is logically equivalent. Justify your answer using truth tables and include a few words of explanation.
\[
(p\vee q)\vee(p\wedge r) \text{ and } (p \vee q)\wedge r.
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Determine whether the following statement is a tautology or a contradiction. Justify your answer using truth tables and include a few words of explanation.
\[
(p\wedge q)\vee (\neg p \vee (p\wedge \neg q)).
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Determine whether the following statement is a tautology or a contradiction. Justify your answer using truth tables and include a few words of explanation.
\[
((\neg p\wedge q)\wedge(q\wedge r))\wedge\neg q.
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Below, a logical equivalence is derived using Theorem 1.1.1 from Epp's book. Supply a reason for each step (which part of the theorem is used).
\[
\begin{array}{rclrr}
(p\wedge \neg q)\vee(p\wedge q) & \equiv & p\wedge(\neg q \vee q) & \text{by} & \underline{(a)}\\
& \equiv & p\wedge(q \vee \neg q) & \text{by} & \underline{(b)}\\
& \equiv & p\wedge{\bf t} & \text{by} & \underline{(c)}\\
& \equiv & p & \text{by} & \underline{(d)}
\end{array}
\]
Therefore, $(p\wedge \neg q)\vee(p\wedge q)\equiv p$.
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Use Theorem 1.1.1 from Epp to verify the logical equivalence. Give a reason for each step (which part of the theorem is used).
\[
(p\wedge(\neg(\neg p \vee q)))\vee (p\wedge q) \equiv p.
\]
\end{ex}
\begin{proof}[Solution]
\end{proof}
\end{document}