ASSIGNMENT.TEX

\documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage{tabular} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \begin{document} \section{Chapter 1 solution} \\ 1) \\ Construct a truth table for each of these compound propositions \begin{description} \item[a)] (p $\wedge$ q) $\rightarrow$ $\neg$q \begin{center} \begin{tabular}{ |c|c|c|c|c| } \hline p & q & $\neg$q & p$\wedge$q & (p$\wedge$q) $\rightarrow$ $\neg$q \\ \hline T & T & F & T & F \\ \hline T & F & T & F & T \\ \hline F & T & F & F & T \\ \hline F & F & T & F & T \\ \hline \end{tabular} \end{center} \item[b)] (p$\vee$r) $\rightarrow$ (r$\vee\neg$p) \begin{center} \begin{tabular}{ |c|c|c|c|c|c| } \hline p & r & $\neg$p & p$\vee$r & r$\vee\neg$p & (p$\vee$r) $\rightarrow$ (r$\vee\neg$p) \\ \hline T & T & F & T & T & T\\ \hline T & F & F & T & F & F\\ \hline F & T & T & T & T & T\\ \hline F & F & T & F & T & T\\ \hline \end{tabular} \end{center} \item[c)] (p$\rightarrow$q) $\vee$ (q$\rightarrow$p) \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline p & q & p$\rightarrow$q & q$\rightarrow$p & (p$\rightarrow$q)$\vee$(q$\rightarrow$p) \\ \hline T & T & T & T & T \\ \hline T & F & F & T & T \\ \hline F & T & T & F & T \\ \hline F & F & T & T & T \\ \hline \end{tabular} \end{center} \item[d)] (p$\vee\neg$q) $\wedge$ ($\neg$p$\vee$q) \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline p & q & $\neg$p & $\neg$q & p$\vee\neg$q & $\neg$p$\vee$q & (p$\vee\neg$q) $\wedge$ ($\neg$p$\vee$q) \\ \hline T & T & F & F & T & T & T \\ \hline T & F & F & T & T & F & F \\ \hline F & T & T & F & F & T & F \\ \hline F & F & T & T & T & T & T \\ \hline \end{tabular} \end{center} \item[e)] (p$\rightarrow\neg$q) $\vee$ (q$\rightarrow\neg$p) \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline p & q & $\neg$p & $\neg$q & p$\rightarrow\neg$q & q$\rightarrow\neg$p & (p$\rightarrow\neg$q) $\vee$ (q$\rightarrow\neg$p) \\ \hline T & T & F & F & F & F & F \\ \hline T & F & F & T & T & T & T \\ \hline F & T & T & F & T & F & T \\ \hline F & F & T & T & T & T & T \\ \hline \end{tabular} \end{center} \item[f)] $\neg$($\neg$p$\wedge\neg$q) \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline p & q & $\neg$p & $\neg$q & $\neg$p$\wedge\neg$q & $\neg$($\neg$p$\wedge\neg$q) \\ \hline T & T & F & F & F & T \\ \hline T & F & F & T & F & T \\ \hline F & T & T & F & F & T \\ \hline F & F & T & T & T & F \\ \hline \end{tabular} \end{center} \item[g)](p$\vee$q) $\rightarrow$ (p$\oplus$q) \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline p & q & p$\vee$q & p$\oplus$q & (p$\vee$q) $\rightarrow$ (p$\oplus$q) \\ \hline T & T & T & F & F \\ \hline T & F & T & T & T \\ \hline F & T & T & T & T \\ \hline F & F & F & F & T \\ \hline \end{tabular} \end{center} \item[h)] (p$\wedge$q) $\vee$ (r$\oplus$q) \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline p & q & r & p$\wedge$q & r$\oplus$q & (p$\wedge$q) $\vee$ (r$\oplus$q) \\ \hline T & T & T & T & F & F \\ \hline T & T & F & T & T & T \\ \hline T & F & T & F & T & T \\ \hline T & F & F & F & F & T \\ \hline F & T & T & F & F & T \\ \hline F & T & F & F & T & T \\ \hline F & F & T & F & T & T \\ \hline F & F & F & F & F & T \\ \hline \end{tabular} \end{center} 2) Negation:\\ a) Steve does not has more than 100 GB free disk space on his laptop.\\ b) Zach does not blocks e-mails and texts from Jennifer.\\ c) 7.11.13 $\ne$ 999.\\ d) Diane did not ride her bicycle 100 miles on Sunday.\\ 3)\\ a) x=2\\ b) x=1\\ c) x=2\\ d) x=2\\ e) x=2\\ 4)\\ a) converse:"I will ski tomorrow only if it snows today"\\ Contrapositive:"If i don't ski tomorrow, then it will not have snowed today."\\ b) Converse:"I come to class only if there going to be a quiz"\\ Contrapositive:"If i don't come to class, then there won't be a quiz"\\ c) Converse:"A positive integer is a prime if it has no divisors other than 1 and itself" Contrapositive:"If a positive integer has a divisor other than 1 and itself, then it is not prime"\\ 5)\\ Let p, q and r be the propositions \\ p : You have the flu. \\ q : You miss the final examination. \\ r : You pass the course. \\ Express each of these propositions as an English sentence. \item[a)] p $\rightarrow$ q \\ You will miss the final examination if you have the flu. \item[b)] $\neg$q $\leftrightarrow$ r \\ If you do not miss the final examination then you will pass the course, and conversely. \item[c)] q $\rightarrow$ $\neg$r \\ You will not pass the course if you miss the final examination. \item[d)] p $\vee$ q $\vee$ r \\ You have the flu or you miss the final examination or you pass the course. \item[e)] (p $\rightarrow$ $\neg$r) $\vee$ (q $\rightarrow$ $\neg$r) \\ You will not pass the course if you have the flu or you miss the final examination. \item[f)] (p $\wedge$ q) $\vee$ ($\neg$q $\wedge$ r) \\ You have the flu and you miss the final examination or you do not miss the final examination and you pass the course.\\ 6)\\ a) r $\wedge$$\neg$q\\ b) p $\wedge$ q $\wedge$ r\\ c) r $\rightarrow$ p\\ d) p $\wedge$ $\neg$q $\wedge$ r\\ e) r $\rightarrow$ (p $\wedge$ q) \\ f) r $\leftrightarrow$ (q$\vee$p)\\ 7)\\ a)11000\\ b)01101\\ 8) parts (a) and (b)\\ \begin{table} \begin{tabular}{ |c| |c| |c| |c| |c| |c| } \hline p & q & p$\wedge$q & (p$\wedge$q)$\rightarrow$p & p$\vee$q & p$\rightarrow$(p$\vee$q) \\ \hline T & T & T & T & T & T \\ \hline T & F & F & T & T & T \\ \hline F & T & F & T & T & T \\ \hline F & F & F & T & F & T \\ \end{tabular} \end{table}\\ parts (c) and (d)\\ \begin{table} \begin{tabular}{ |c| |c| |c| |c| |c| |c| |c| } \hline p & q & $\neg$p & p$\rightarrow$q & $\neg$p$\rightarrow$(p$\rightarrow$q) & p$\wedge$q & (p$\wedge$q)$\rightarrow$(p$\rightarrow$q) \\ \hline T & T & F & T & T & T & T \\ \hline T & F & F & F & T & F & T \\ \hline F & T & T & T & T & F & T \\ \hline F & F & T & T & T & F & T \\ \end{tabular} \end{table}\\ parts (e) and (f)\\ \begin{table} \begin{tabular}{ |c| |c| |c| |c| |c| |c| |c| } \hline p & q & p$\rightarrow$q & $\neg$(p$\rightarrow$q) & $\neg$(p$\rightarrow$q)$\rightarrow$p & $\neg$q & $\neg$(p$\rightarrow$q)$\rightarrow$$\neg$q \\ \hline T & T & T & F & T & F & T \\ \hline T & F & F & T & T & T & T \\ \hline F & T & T & F & T & F & T \\ \hline F & F & T & F & T & T & T \\ \end{tabular} \end{table}\\ parts (g) \begin{table} \begin{tabular}{ |c| |c| |c| |c| |c| |c| } \hline p & q & $\neg$p & p$\vee$q & [$\neg$p$\wedge$(p$\vee$q)] & [$\neg$p$\wedge$(p$\vee$q)]$\rightarrow$q \\ \hline T & T & F & T & F & T \\ \hline T & F & F & T & F & T \\ \hline F & T & T & T & T & T \\ \hline F & F & T & F & F & T \\ \end{tabular} \end{table} parts (h) \begin{table} \begin{tabular}{ |c| |c| |c| |c| |c| |c| |c| |c| } \hline p & q & r & p$\rightarrow$q & p$\rightarrow$r & q$\rightarrow$r & [(p$\rightarrow$q)$\wedge$(q$\rightarrow$r)] & [(p$\rightarrow$q)$\wedge$(q$\rightarrow$r)]$\rightarrow$(p$\rightarrow$r) \\ \hline T & T & T & T & T & T & T & T\\ \hline T & F & F & F & F & T & F & T\\ \hline T & F & T & F & T & T & F & T\\ \hline T & T & F & T & F & F & F & T\\ \hline F & T & F & T & T & F & F & T\\ \hline F & T & T & T & T & T & T & T\\ \hline F & F & F & T & T & T & T & T\\ \hline F & F & T & T & T & T & T & T\\ \end{tabular} \end{table}\\ 9)\\ Show that these compound propositionals are logically equivalent. \item[a)] $\neg$(p $\leftrightarrow$ q) and $\neg$p $\leftrightarrow$ q \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline p & q & $\neg$p & p$\leftrightarrow$q & $\neg$(p$\leftrightarrow$q) & $\neg$p$\leftrightarrow$q \\ \hline T & T & F & T & F & F \\ \hline T & F & F & F & T & T \\ \hline F & T & T & F & T & T \\