% If you are working from this template to just write up your own
% solutions, I'm trying to keep it simple for you and there are only a
% few places you need to edit. On line 43 there's a place where the
% phrase "YOUR NAME HERE" appears; change it to your actual
% name. Several places below there are places where the phrase "YOUR
% ANSWER HERE" appears, and directly below (or replacing) those lines
% are good places to put your answers.
%
% One important thing you need to do, however (due to the complicated
% way I use a single form to produce both the question sheets and
% solution writeups) is to make sure that you keep the word 'solution'
% in your filename, or else your answers won't be printed at all.
\documentclass[12pt]{article}
\usepackage[margin=0.7in,dvips]{geometry}
\usepackage{fancyhdr,lastpage}
\usepackage{amsmath,amsfonts}
\usepackage{tikz}
\usepackage{substr}
\IfSubStringInString{\detokenize{solution}}{\jobname}{%
% Solutions
\newcommand\qfill{}
\newcommand\qpage{}
\newcommand\qonly[1]{}
\newcommand\answer[1]{#1}
\newcommand\question[1]{{\em #1}}
\newcommand\rubric[1]{}
}
{%
% Question sheets
\newcommand\qfill{\vfill}
\newcommand\qpage{\newpage}
\newcommand\qonly[1]{#1}
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\newcommand\question[1]{#1}
\newcommand\rubric[1]{}
}
\pagestyle{fancy}
\lhead{MATH 311-01}
\chead{Problem Set \#2\answer{ solutions}}
\rhead{\answer{YOUR NAME HERE}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due January 26, 2017}}
% The command below redefines \neg (the logical negation character)
% to be a big tilde, in line with the usage in the book. Comment it
% out if you prefer the more traditional negation symbol.
\renewcommand*{\neg}{\mathord{\sim}}
\begin{document}
\begin{enumerate}
\item \question{Produce truth tables for each of the following statements; if the truth table describes a tautology or contradiction, say so.}
\begin{enumerate}
\item \question{$(P\vee Q)\rightarrow\neg Q$.}
\answer{
% Add as many "c"s separated by "|"s as there should be columns
% in the table; as it stands now, this is a table with two
% columns, then a double line, then 3 more columns.
\begin{tabular}{|c|c||c|c|c|}\hline
% The first row will be the header; there are some sample
% entries here to get you started, but they probably aren't
% "right" for this question.
$P$&$Q$&$P\vee Q$&$\neg P$&$Q\rightarrow P$\\\hline\hline
% Henceforth, the rows are the truth values in the table. Two
% rows with bogus data ad one blank row appear below; you can
% copy and paste the blank row, if you like, to get more.
T&T&T&F&T\\\hline
F&F&F&T&T\\\hline
&&&&\\\hline
\end{tabular}
}
\item \question{$[(P\rightarrow Q)\wedge (Q\rightarrow R)]\rightarrow (P\rightarrow R)$.}
\answer{
% Add as many "c"s separated by "|"s as there should be columns
% in the table; as it stands now, this is a table with two
% columns, then a double line, then 3 more columns.
\begin{tabular}{|c|c||c|c|c|}\hline
% The first row will be the header; there are some sample
% entries here to get you started, but they probably aren't
% "right" for this question.
$P$&$Q$&$P\vee Q$&$\neg P$&$Q\rightarrow P$\\\hline\hline
% Henceforth, the rows are the truth values in the table. Two
% rows with bogus data ad one blank row appear below; you can
% copy and paste the blank row, if you like, to get more.
T&T&T&F&T\\\hline
F&F&F&T&T\\\hline
&&&&\\\hline
\end{tabular}
}
\end{enumerate}
\item \question{Demonstrate, using a truth table or any other
technique, that the statements $(P\vee Q)\wedge R$ and $(P\wedge
R)\vee (Q\wedge R)$ are logically equivalent.}
\item \question{The following four propositions are taken from a work
of Lewis Carroll's. Translate each into a symbolic quantified
statement (i.e. a statement using either the universal
quantifier $\forall$ or the existential quantifier $\exists$) using the
following names for sets and open statements:
\begin{itemize}
\item $U$: the set of all books
\item $B(x)$: the statement ``book $x$ is bound.''
\item $R(x)$: the statement ``book $x$ is recommended for reading.''
\item $H(x)$: the statement ``book $x$ is healthy in tone.''
\item $W(x)$: the statement ``book $x$ is well-written.''
\end{itemize}}
\begin{enumerate}
\item The only books that I do not recommend for
reading are unhealthy in tone.
\answer{
%YOUR ANSWER HERE
}
\item The bound books are all well-written.
\answer{
%YOUR ANSWER HERE
}
\item All the romances are healthy in tone.
\answer{
%YOUR ANSWER HERE
}
\item I do not recommend you to read any of the unbound books.
\answer{
%YOUR ANSWER HERE
}
\end{enumerate}
\item \question{Based on the quantified statements in the previous
question, what conclusions could you reach if you knew that the
book ``The Old Man's Comforts'' is healthy in tone?}
\end{enumerate}
\end{document}