From 89862ae6a0554870a7708ae73112f86d2d21fc8d Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Thu, 10 Feb 2022 01:12:45 -0500
Subject: new teachers, new work

---
 howard/hw1.tex | 248 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 248 insertions(+)
 create mode 100644 howard/hw1.tex

(limited to 'howard/hw1.tex')

diff --git a/howard/hw1.tex b/howard/hw1.tex
new file mode 100644
index 0000000..b9731b2
--- /dev/null
+++ b/howard/hw1.tex
@@ -0,0 +1,248 @@
+\newcount\indentlevel
+\newcount\itno
+\def\reset{\itno=1}\reset
+\def\afterstartlist{\advance\leftskip by .5in\par\advance\leftskip by -.5in}
+\def\startlist{\par\advance\indentlevel by 1\advance\leftskip by .5in\reset
+\aftergroup\afterstartlist}
+\def\alph#1{\ifcase #1\or a\or b\or c\or d\or e\or f\or g\or h\or
+    i\or j\or k\or l\or m\or n\or o\or p\or q\or r\or
+    s\or t\or u\or v\or w\or x\or y\or z\fi}
+\def\li#1\par{\medskip\item{\ifcase\indentlevel \number\itno.\or
+                                  \alph\itno)\else
+                                  (\number\itno)\fi
+                           }%
+         #1\smallskip\advance\itno by 1}
+\def\hline{\noalign{\hrule}}
+\let\impl\rightarrow
+\newskip\tableskip
+\tableskip=10pt plus 10pt
+
+\li Evaluate each of the conditional statements to true or false
+
+{\startlist
+    \li If $1+2=4,$ then $9+0= -9$
+
+        With $p$ $1+2=4$ and $q$ $9+0= -9,$ this statement is $p\impl
+        q.$
+        $1+2\neq 4,$ so this conditional is vacuously true.
+
+    \li $13 > 19$ only if 13 is prime
+
+        With $p$ 13 is prime and $q$ $13 > 19,$ this statement is
+        $p\impl q,$ and since $p$ is true and $q$ is false, this
+        statement is false.
+
+    \li Horses can fly whenever horses cannot fly
+
+        With $p$ ``horses cannot fly'' and $q$ ``horses can fly,'' this
+        is equivalent to $p\impl q.$ Since horses cannot fly, $p$ is
+        true and $q$ is false, so the statement is false.
+
+    \li $3\cdot3 = 9,$ if $9+9 = 18$
+
+        With $p$ $9+9 = 18$ and $q$ $3\cdot3=9,$ this statement is
+        equivalent to $p\impl q,$ and since both are true, this
+        statement is true.
+}
+
+
+\li Let $p$ and $q$ be propositions, where $p$ is the statement ``It is
+snowing outside,'' and $q$ is the statement ``It is June.'' Express each
+of the following propositions as an English sentence.
+
+{\startlist
+    \li $p\impl q.$
+
+    ``If it is snowing outside, it is June.''
+
+    \li $\lnot p \land q.$
+
+    ``It is not snowing outside, and it is June.''
+
+    \li $\lnot p \lor (p\land q).$
+
+    ``It is not snowing outside, or it is snowing outside in June.''
+}
+
+\li Consider the statement: ``If the TAs make the homework too hard,
+then the students will be sad.'' Write the converse, contrapositive, and
+inverse of the statement. Don't worry about the grammar/tense, we just
+want to see the correct idea.
+
+{\startlist
+    \li Converse
+
+    ``If the students will be sad, the TAs make the homework too hard.''
+
+    \li Contrapositive
+
+    ``If the students aren't sad, the TAs didn't make the homework too
+    hard.''
+
+    \li Inverse
+
+    ``If the TAs don't make the homework too hard, the students will be
+    happy.''
+}
+
+\li How many rows appear in a truth table for each of these compound
+propositions?
+
+{\startlist
+    \li $p\land q$
+
+    The number of rows is $2^v$ where v is the number of variables in
+    the expression. Therefore, this expression will have $2^2 = 4$ rows.
+
+    \li $\lnot p \impl (p\impl q)$
+
+    This still only has two variables, so it will have $2^2 = 4$ rows.
+
+    \li $(\lnot p\land q\land s)\lor(p\land\lnot q\land s)\lor(p\land
+    q\land\lnot s)$
+
+    This has three variables, so it will have $2^3 = 8$ rows.
+}
+
+\li Using the following propositions, translate the sentence ``You
+cannot see the movie if you are not over 18 years old and you do not
+have the permission of a parent'' to a compound proposition.
+
+$m := \hbox{``You can see the movie''}$
+
+$e := \hbox{``You are over 18 years old''}$
+
+$p := \hbox{``You have the permission of a parent''}$
+
+This is $(\lnot p\land\lnot e)\impl \lnot m.$
+
+\li Construct truth tables for the following propositions. Include all
+intermediate columns to receive full credit for each table.
+
+{\startlist
+    \li $p\lor q\land\lnot p$
+
+    \leavevmode
+    \halign{&\vrule\strut#&#\tabskip\tableskip&#&#\tabskip0pt\cr\hline
+
+        &&$p$&&&&$q$&&&&$\lnot p$&&&&$q\land\lnot p$&&&&
+        $p\lor q\land\lnot p$&&\cr\hline
+        &&T&&&&T&&&&F&&&&F&&&&T&&\cr\hline
+        &&T&&&&F&&&&F&&&&F&&&&T&&\cr\hline
+        &&F&&&&T&&&&T&&&&T&&&&T&&\cr\hline
+        &&F&&&&F&&&&T&&&&F&&&&F&&\cr\hline
+        }
+
+    \li $(p\lor\lnot q) \impl q$
+
+    \leavevmode
+    \halign{&\vrule\strut#&#\tabskip\tableskip&#&#\tabskip0pt\cr\hline
+
+        &&$p$&&&&$q$&&&&$\lnot q$&&&&$p\lor\lnot q$&&&&
+        $(p\lor\lnot q)\impl q$&&\cr\hline
+        &&T&&&&T&&&&F&&&&T&&&&T&&\cr\hline
+        &&T&&&&F&&&&T&&&&T&&&&F&&\cr\hline
+        &&F&&&&T&&&&F&&&&F&&&&T&&\cr\hline
+        &&F&&&&F&&&&T&&&&T&&&&F&&\cr\hline
+        }
+
+    \li $(\lnot q\impl\lnot q) \iff (\lnot p \impl \lnot q)$
+
+    \leavevmode
+    \halign{&\vrule\strut#&#\tabskip\tableskip&#&#\tabskip0pt\cr\hline
+
+        &&$p$&&&&$q$&&&&$\lnot p$&&&&$\lnot q$&&&&$\lnot q\impl\lnot q$&&&&
+        $\lnot p\impl\lnot q$&&&&
+        $(\lnot q\impl\lnot q)\iff(\lnot p\impl\lnot q)$&&\cr\hline
+        &&T&&&&T&&&&F&&&&F&&&&T&&&&T&&&&T&&\cr\hline
+        &&T&&&&F&&&&F&&&&T&&&&T&&&&T&&&&T&&\cr\hline
+        &&F&&&&T&&&&T&&&&F&&&&T&&&&F&&&&F&&\cr\hline
+        &&F&&&&F&&&&T&&&&T&&&&T&&&&T&&&&T&&\cr\hline
+        }
+}
+
+\li There is a spaceship where every passenger has exactly one role.
+Each passenger can either be a Crewmate or an Imposter. A Crewmate
+always tells the truth, and an Imposter always lies. For each question,
+determine the role of Person A and the role of Person B, or write
+``Cannot be determined'' for that person if there is not enough
+information. Explain your reasoning for full credit! (You can use a
+truth table or just plain English to explain.)
+
+{\startlist
+    \li Person A says ``I am a Crewmate, or B is a Crewmate,'' and
+    Person B says ``A is a Crewmate if I am an Imposter.''
+
+    If and only if A is a crewmate, their statement is true. Similarly,
+    if and only if B is a crewmate, their statement is true. We will use
+    this principle for all of the problems.
+
+    In this problem, if both are crewmates, both statements are true (B
+    is not an imposter, so their statement is vacuously true, and A is a
+    crewmate or B is a crewmate).
+
+    However, if both are imposters, both statements are false (neither
+    is a crewmate and A is not a crewmate even though B is an imposter).
+    This means we can't figure out any information.
+
+    \li Person A says ``I am an Imposter, and Person B is a Crewmate,''
+    and Person B says nothing.
+
+    Both must be imposters. If A is a crewmate, their statement ``I am
+    an imposter'' is a lie, so they are an imposter by contradiction.
+    Since A is an an imposter, this statement must be a lie, so ``Person
+    B is a crewmate'' is false, and B is an imposter.
+
+    \li Person A says ``Both Person B and I are Imposters,'' and Person
+    B says ``At least one of us is a Crewmate.''
+
+    By a similar logic as b, ``I am an imposter'' would be a lie if A
+    were a crewmate, so A is an imposter. And since that statement is
+    true, ``Person B is an imposter'' must be a lie, and Person B is a
+    crewmate. This makes B's statement true, validating our view.
+}
+
+\li Show that $(p\impl q)\lor\lnot p \equiv p\impl q$ using logical
+equivalences. Cite the laws of equivalences used to reach each step.
+
+\leavevmode
+\goodbreak
+\halign{\vrule\strut#\tabskip\tableskip&#\hfil&#\hfil&#\tabskip0pt&#\vrule\cr\hline
+&$(p\impl q)\lor\lnot p$&Given&&\cr
+&$(\lnot p\lor q)\lor\lnot p$&Conditional Identity&&\cr
+&$(q\lor\lnot p)\lor\lnot p$&Commutative Law&&\cr
+&$q\lor(\lnot p\lor\lnot p)$&Associative Law&&\cr
+&$q\lor\lnot p$&Idempotent Law&&\cr
+&$p\impl q$&Conditional Identity&&\cr
+\hline
+}
+
+\li Show that $\lnot ((p\land q)\lor p)\impl \lnot p$ is a tautology
+using:
+
+{\startlist
+    \li a truth table (include all intermediate columns)
+
+    \halign{&\vrule\strut#&#\tabskip\tableskip&#&#\tabskip0pt\cr\hline
+
+        &&$p$&&&&$q$&&&&$p\land q$&&&&$(p\land q)\lor p$&&&&
+        $\lnot((p\land q)\lor p)$&&&&
+        $\lnot p$&&&&$\lnot ((p\land q)\lor p)\impl \lnot p$&&\cr\hline
+        &&T&&&&T&&&&T&&&&T&&&&F&&&&F&&&&T&&\cr\hline
+        &&T&&&&F&&&&F&&&&T&&&&F&&&&F&&&&T&&\cr\hline
+        &&F&&&&T&&&&F&&&&F&&&&T&&&&T&&&&T&&\cr\hline
+        &&F&&&&F&&&&F&&&&F&&&&T&&&&T&&&&T&&\cr\hline
+    }
+
+    \li logical equivalences (cite the laws of equivalence used to reach
+    each step)
+
+    \halign{\vrule\strut#\tabskip\tableskip&#\hfil&#\hfil&#\tabskip0pt&#\vrule\cr\hline
+    &$\lnot ((p\land q)\lor p)\impl \lnot p$&Given&&\cr
+    &$\lnot p\impl \lnot p$&Absorption Law&&\cr
+    &$p\lor\lnot p$&Conditional Identity&&\cr
+    &$T$&Complement Law&&\cr
+    \hline
+    }
+}
+\bye
-- 
cgit 

