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diff --git a/hw03/HW03DanielShchur.pdf b/hw03/HW03DanielShchur.pdf
index addbc89726cefcb4cf0e028a3e7746c696877964..2dcfc278dc4a3682de7b4ab80182a9f035607f5e 100755
Binary files a/hw03/HW03DanielShchur.pdf and b/hw03/HW03DanielShchur.pdf differ
diff --git a/hw03/HW03DanielShchur.tex b/hw03/HW03DanielShchur.tex
index ab34c68368ae0ac7581cbc7279ade3452245079f..0bbe661dc7493e4832f6742d996f252dd5fcfffe 100644
--- a/hw03/HW03DanielShchur.tex
+++ b/hw03/HW03DanielShchur.tex
@@ -59,22 +59,22 @@
\item[a)]$\exists x (x^3 = -1)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[b)]$\exists x (x^4 < x^2)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[c)]$\forall x ((-x)^2 = x^2)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[d)]$\forall x (2x>x)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False, if the number is negative, $2x<x$. Ex: $x=-2, 2(-2) = -4, 1(-2) = -2, -4 < -2$
\end{enumerate}
\end{enumerate}
@@ -85,27 +85,27 @@
\item[a)]Something is not in the correct place.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is an item and $P(x)$ denotes correct placement: $\exists x (\neg P(x))$
\end{enumerate}
\item[b)]All tools are in the correct place and are in excellent condition.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a tool, $P(x)$ denotes correct placement, and $C(x)$ denotes excellent condition: $\forall x (P(x) \wedge C(x))$
\end{enumerate}
\item[c)]Everything is in the correct place and in excellent condition.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is an item, $P(x)$ denotes correct placement, and $C(x)$ denotes excellent condition: $\forall x (P(x) \wedge C(x))$
\end{enumerate}
\item[d)]Nothing is in the correct place and is in excellent condition.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is an item, $P(x)$ denotes correct placement, and $C(x)$ denotes excellent condition: $\neg \forall x (P(x) \wedge C(x))$
\end{enumerate}
\item[e)]One of your tools is not in the correct place, but it is in excellent condition.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a tool, $P(x)$ denotes correct placement, and $C(x)$ denotes excellent condition: $\exists x (\neg P(x) \wedge C(x))$
\end{enumerate}
\end{enumerate}
@@ -116,27 +116,52 @@
\item[a)]All dogs have fleas.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a dog and $F(x)$ denotes ``has fleas.''
+ \begin{enumerate}
+ \item[1)] $\forall x (F(x))$
+ \item[2)] $\neg \forall x (F(x)) \equiv \exists x (\neg F(x))$
+ \item[3)] All dogs do not have fleas.
+ \end{enumerate}
\end{enumerate}
\item[b)]There is a horse that can add.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a horse and $A(x)$ denotes ``can add.''
+ \begin{enumerate}
+ \item[1)] $\exists x (A(x))$
+ \item[2)] $\neg \exists x (A(x)) \equiv \forall x (\neg A(x))$
+ \item[3)] There are no horses that can add.
+ \end{enumerate}
\end{enumerate}
\item[c)]Every koala can climb.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a koala and $C(x)$ denotes ``can climb.''
+ \begin{enumerate}
+ \item[1)] $\forall x (C(x))$
+ \item[2)] $\neg \forall x (C(x)) \equiv \exists x (\neg C(x))$
+ \item[3)] Not every Koala can climb.
+ \end{enumerate}
\end{enumerate}
\item[d)]No monkey can speak French.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a monkey and $S(x)$ denotes ``can speak French.''
+ \begin{enumerate}
+ \item[1)] $\forall x (\neg S(x))$
+ \item[2)] $\neg \forall x (\neg S(x)) \equiv \exists x (S(x))$
+ \item[3)] There is a monkey that can speak French.
+ \end{enumerate}
\end{enumerate}
\item[e)]There exists a pig that can swim and catch fish.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a pig, $S(x)$ denotes ``can swim,'' and $C(x)$ denotes ``catch fish.''
+ \begin{enumerate}
+ \item[1)] $\exists x (S(x) \wedge C(x))$
+ \item[2)] $\neg \exists x (S(x) \wedge C(x)) \equiv \forall x (\neg S(x) \vee \neg C(x))$
+ \item[3)] There are no pigs that can swim or catch fish.
+ \end{enumerate}
\end{enumerate}
\end{enumerate}
@@ -148,38 +173,52 @@
\item[b)]All Swedish movies are serious.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a Swedish movie and $S(x)$ denotes ``are serious.''
+ \begin{enumerate}
+ \item[1)]$\exists x (\neg S(x))$
+ \item[2)]There is a Swedish movie that is not serious.
+ \end{enumerate}
\end{enumerate}
\item[c)]No one can keep a secret.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a person and $K(x)$ denotes ``can keep a secret.''
+ \begin{enumerate}
+ \item[1)]$\exists x(K(x))$
+ \item[2)]There is a person that can keep a secret.
+ \end{enumerate}
\end{enumerate}
\item[d)]There is someone in this class who does not have a good attitude.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Where $x$ is a person in this class and $A(x)$ denotes ``has a good attitude.''
+ \begin{enumerate}
+ \item[1)] $\forall x (A(x))$
+ \item[2)] Everyone in this class has a good attitude.
+ \end{enumerate}
\end{enumerate}
\end{enumerate}
+\newpage
+
\problem (\#36 S1.4 Pg55) Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all real numbers.
\begin{enumerate}
\item[a)]$\forall x (x^2 \neq x)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False, if $x=1$, then $(1)^2 \equiv (1)$
\end{enumerate}
\item[b)]$\forall x (x^2 \neq 2)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer When $x = \sqrt 2$, then $(\sqrt 2)^2 \equiv 2$
\end{enumerate}
\item[c)]$\forall x (|x| > 0)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False, if $x=0$, then $|0| \equiv 0$
\end{enumerate}
\end{enumerate}
@@ -190,12 +229,12 @@
\item[b)]$\forall x(S(x$, malfunctioning$) \vee S(x$, diagnostic))
\begin{enumerate}
- \item[]\answer
+ \item[]\answer All systems are either malfunctioning or are running diagnostics.
\end{enumerate}
\item[e)]$\forall x \neg S(x$, working)
\begin{enumerate}
- \item[]\answer
+ \item[]\answer All systems are not working.
\end{enumerate}
\end{enumerate}
@@ -206,17 +245,17 @@
\item[a)]$C$(Randy Goldberg, CS 252)
\begin{enumerate}
- \item[]\answer
+ \item[]\answer Randy Goldberg is enrolled in the class CS 252.
\end{enumerate}
\item[b)]$\exists x C(x$, Math 695)
\begin{enumerate}
- \item[]\answer
+ \item[]\answer There is a student who is enrolled in the class Math 695.
\end{enumerate}
\item[d)]$\exists x (C(x$, Math 222$) \wedge C(x$, CS 252)
\begin{enumerate}
- \item[]\answer
+ \item[]\answer There is a student who is enrolled in Math 222 and CS 252.
\end{enumerate}
\end{enumerate}
@@ -227,27 +266,27 @@
\item[e)]Everyone can be fooled by somebody.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\exists x \forall y (F(x, y))$
\end{enumerate}
\item[g)]Nancy can fool exactly two people.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\exists !x,!y (F($Nancy$,x) \wedge F($Nancy$,y)$
\end{enumerate}
\item[h)]There is exactly one person whom everybody can fool.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\forall x \exists !y (F(x, y))$
\end{enumerate}
\item[i)]No one can fool himself or herself.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\forall x (\neg F(x, x))$
\end{enumerate}
\item[j)]There is someone who can fool exactly one person besides himself or herself.
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\exists !x,!y (F(x, y) \wedge F(x,x))$
\end{enumerate}
\end{enumerate}
@@ -258,27 +297,27 @@
\item[a)]$\forall x \exists y (x^2 = y)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[c)]$\exists x \forall y (xy = 0)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[e)]$\forall x(x \neq 0 \rightarrow \exists y(xy = 1))$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[g)]$\forall x \exists y (x + y = 1)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer True
\end{enumerate}
\item[i)]$\forall x \exists y (x + y = 2 \wedge 2x - y = 1)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False
\end{enumerate}
\end{enumerate}
@@ -287,19 +326,21 @@
\begin{enumerate}
-\item[b)]$\neg \forall x \exists P(x, y)$
+\item[b)]$\neg \forall x \exists y (P(x, y))$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\exists x \forall y (\neg P(x, y))$
\end{enumerate}
\item[d)]$\neg \exists y (\exists x R(x, y) \vee \forall x S(x, y))$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\forall y (\forall x \neg R(x, y) \wedge \exists x \neg S(x, y))$
\end{enumerate}
+
+ \newpage
\item[e)]$\neg \exists y (\forall x \exists z T(x, y, z) \vee \exists x \forall z U(x, y, z))$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer $\forall y (\exists x \forall z \neg T(x,y,z) \wedge \forall x \exists z \neg U(x, y, z))$
\end{enumerate}
\end{enumerate}
@@ -310,17 +351,17 @@
\item[a)]$\forall x \exists y(x = 1/y)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False, $1/0$ is undefined and therefore makes $x = 0$ impossible.
\end{enumerate}
\item[b)]$\forall x \exists y(y^2 - x < 100)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False, if $x<-100$ then there is no number $y$ that will get to $<100$
\end{enumerate}
\item[c)]$\forall x \forall y(x^2 \neq y^3)$
\begin{enumerate}
- \item[]\answer
+ \item[]\answer False, when $x=1 \wedge y=1, x^2 = y^3$
\end{enumerate}
\end{enumerate}
@@ -329,4 +370,18 @@
\answerindent
+\begin{table}[htb]
+ \caption{Problem #48}
+ \centering
+ \begin{tabular}{l l | l}
+ \hline
+ 1. & $\forall x P(x) \vee \forall x Q(x)$ & Original\\
+ 2. & $\equiv \forall x P(x) \vee \forall y Q(y)$ & Redefinition using another variable\\
+ 3. & $\equiv \forall x,y (P(x) \vee Q(y))$ & Since seperate variables, can combine quantifers (Idempotent)\\
+ \hline
+ \multicolumn{3}{c}{$\forall x P(x) \vee \forall x Q(x) \equiv \forall x \forall y (P(x) \vee Q(y))$}\\
+ \hline
+ \end{tabular}
+\end{table}
+
\end{document}
\ No newline at end of file
diff --git a/hw03/HW03DanielShchur.tex~ b/hw03/HW03DanielShchur.tex~
deleted file mode 100644
index fc886fa08a50aaee91eeb0fd1830c980399b024c..0000000000000000000000000000000000000000
--- a/hw03/HW03DanielShchur.tex~
+++ /dev/null
@@ -1,56 +0,0 @@
-\documentclass[12pt,letterpaper]{article}
-
-\usepackage{bm}
-\usepackage{forest}
-\usepackage[letterpaper, margin=.75in]{geometry}
-
-\newcommand{\problem}{\vspace{.5cm}\textbf{Problem:~~}}
-\newcommand{\answer}{\emph{Answer:~~}}
-\newcommand{\ansindent}{\indent\hspace{0.5cm}}
-\newcommand{\answerindent}{\hangindent=.5cm\hangafter=0\answer}
-\newcommand{\ansend}{\vspace{0.25cm}\\}
-
-\setlength{\parindent}{0pt}
-\setlength{\parskip}{0cm}
-\setlength{\intextsep}{.25cm}
-
-\title{CSCE 235H Homework 03}
-\author{Daniel Shchur}
-
-\begin{document}
-
-\maketitle
-
-\problem (\#10 S1.4 Pg53)
-
-\problem (\#14 S1.4 Pg53)
-
-\problem (\#28 S1.4 Pg54)
-
-\problem (\#32 S1.4 Pg55)
-
-\problem (\#34 S1.4 Pg55)
-
-\ansindent\answer
-
-\begin{enumerate}
-
-\item[b]
- \begin{enumerate}
- \item[]\answer
- \end{enumerate}
-
-\item[c]
- \begin{enumerate}
- \item[]\answer
- \end{enumerate}
-
-\item[d]
- \begin{enumerate}
- \item[]\answer
- \end{enumerate}
-
-\end{enumerate}
-
-
-\end{document}
\ No newline at end of file
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old mode 100755
new mode 100644
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rename from hw03/hw03.pdf
rename to hw03/cse235-hw03.pdf
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new file mode 100644
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--- /dev/null
+++ b/hw04/HW04DanielShchur.tex
@@ -0,0 +1,265 @@
+\documentclass[12pt,letterpaper]{article}
+
+\usepackage[table]{xcolor}
+\usepackage{boldline}
+\usepackage{bm}
+\usepackage{forest}
+\usepackage[letterpaper, margin=.75in]{geometry}
+\usepackage{adjustbox}
+
+\newcommand{\problem}{\vspace{.5cm}\textbf{Problem:~~}}
+\newcommand{\problemEnum [1]}{\vspace{.5cm}\textbf{Problem {#1}:~~}}
+\newcommand{\answer}{\emph{Answer:~~}}
+\newcommand{\ansindent}{\indent\hspace{0.5cm}}
+\newcommand{\ansend}{\vspace{0.25cm}\\}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0cm}
+\setlength{\intextsep}{.25cm}
+
+\title{CSCE 235H Homework 04}
+\author{Daniel Shchur}
+
+\begin{document}
+
+\maketitle
+
+\problem (\#4 S1.6 Pg78) What rule of inference is used in each of these arguments?
+
+\begin{enumerate}
+\item[a)]Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials.
+
+ \begin{enumerate}
+ \item[]\answer Simplification Rule
+ \end{enumerate}
+
+ \item[b)]It is either hotter than 100 degrees today or the pollution is dangerous. It is less than 100 degrees outside today. Therefore, the pollution is dangerous.
+
+ \begin{enumerate}
+ \item[]\answer Disjunctive Syllogism / Unit Resolution
+ \end{enumerate}
+
+ \item[c)] Linda is an excellent swimmer. If Linda is an excellent swimmer, then she can work as a lifeguard. Therefore, Linda can work as a lifeguard.
+
+ \begin{enumerate}
+ \item[]\answer Modus Ponens
+ \end{enumerate}
+
+\item[d)]Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or he will be a beach bum.
+
+ \begin{enumerate}
+ \item[]\answer Addition
+ \end{enumerate}
+
+\item[e)]If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.
+
+ \begin{enumerate}
+ \item[]\answer Hypothetical Syllogism / Transitivity
+ \end{enumerate}
+\end{enumerate}
+
+\newpage
+
+\problem (\#10 S1.6 Pg79) For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.
+
+\begin{enumerate}
+\item[a)]``If I play hockey, then I am sore the next day.''\\
+ ``I use the whirlpool if I am sore.''\\
+ ``I did not use the whirlpool.''
+
+ \begin{enumerate}
+ \item[]\answer Modus Ponens gives ``If I play hockey, then I use the whirlpool,'' Modus Tollens gives ``I did not play hockey.''
+ \end{enumerate}
+
+\item[b)]``If I work, it is either sunny or partly sunny.''\\
+ ``I worked last Monday or I worked last Friday.''\\
+ ``It was not sunny on Tuesday.''\\
+ ``It was not partly sunny on Friday.''
+
+ \begin{enumerate}
+ \item[]\answer Modus Tollens gives ``I did not work on Tuesday,'' Modus Ponens gives ``(It was sunny or partly sunny on Monday) or it was sunny on Friday.''
+ \end{enumerate}
+
+\item[c)]``All insects have six legs.''\\
+``Dragonflies are insects''\\
+``Spiders do not have six legs.''\\
+``Spiders eat dragonflies.''
+
+ \begin{enumerate}
+ \item[]\answer Modus Tollens gives ``Spiders are not insects,'' Transitivity gives ``Spiders eat insects.''
+ \end{enumerate}
+
+\item[d)]``Every student has an Internet account.''\\
+``Homer does not have an Internet account.''\\
+``Maggie has an Internet account.''
+
+ \begin{enumerate}
+ \item[]\answer Modus Ponens gives ``Maggie is a student,'' Modus Tollens gives ``Homer is not a student.''
+ \end{enumerate}
+
+\item[e)]``All foods that are healthy to eat do not taste good.''\\
+``Tofu is healthy to eat.''\\
+``You only eat what tastes good.''\\
+``You do not eat tofu.''\\
+``Cheeseburgers are not healthy to eat.''
+
+ \begin{enumerate}
+ \item[]\answer Modus Ponens gives ``Tofu does not taste good,'' Modus Tollens gives ``You only eat unhealthy food.''
+ \end{enumerate}
+\end{enumerate}
+
+\newpage
+
+\problemEnum[A] Prove the statement below. You may use equivalence rules or rewrite both the sentences in conjunctive Normal Form but you may not use truth tables.\\
+$(p \wedge q) \rightarrow r \equiv p \rightarrow (q \rightarrow r)$\vspace{0.5cm}
+
+\answer\vspace{0.2cm}
+
+\begin{table}[htb]
+ \centering
+ \begin{tabular}{l l | l}
+ \hline
+ 1. & $p \rightarrow (q \rightarrow r)$ & Premise\\
+ 2. & $\neg p \vee (q \rightarrow r)$ & Implication Law\\
+ 3. & $\neg p \vee (\neg q \vee r)$ & Implication Law\\
+ 4. & $(\neg p \vee \neg q) \vee r$ & Associative Law\\
+ 5. & $\neg (p \wedge q) \vee r$ & DeMorgan's Law\\
+ 6. & $(p \wedge q) \rightarrow r$ & Implication Law\\
+ \hline
+ \multicolumn{3}{c}{$(p \wedge q) \rightarrow r \equiv p \rightarrow (q \rightarrow r)$}\\
+ \hline
+ \end{tabular}
+\end{table}
+
+\problemEnum[B]
+
+\begin{enumerate}
+\item[1.]Rewrite the following axioms in Conjunctive Normal Form:\\
+ $(p \wedge t) \rightarrow (r \vee s), q \rightarrow (u \wedge t), u\rightarrow p,$ and $\neg s$\\
+
+ \answer
+ \begin{table}[htb]
+ \centering
+ \tabcolsep=.06cm
+ \begin{tabular}{l l l | l l | l l | l l}
+ \hline
+ 1. & $(p \wedge t) \rightarrow (r \vee s)$ & Premise & $q \rightarrow (u \wedge t)$ & Premise & $u\rightarrow p$ & Premise & $\neg s$ & CNF\\
+ 2. & $\neg (p \wedge t) \vee (r \vee s)$ & Implication & $\neg q \vee (u \wedge t)$ & Implication & $\neg u \vee p$ & Implication & $\neg s$ & CNF\\
+ 3. & $(\neg p \vee \neg t) \vee (r \vee s)$ & DeMorgan's & $(\neg q \vee u) \wedge (\neg q \vee t)$ & Distributive & $\neg u \vee p$ & CNF & $\neg s$ & CNF\\
+ 4. & $(\neg p \vee \neg t \vee r \vee s)$ & Associative & $(\neg q \vee u) \wedge (\neg q \vee t)$ & CNF & $\neg u \vee p$ & CNF & $\neg s$ & CNF\\
+ \hline
+ \multicolumn{9}{c}{$(\neg p \vee \neg t \vee r \vee s) \wedge (\neg q \vee u) \wedge (\neg q \vee t) \wedge (\neg u \vee p) \wedge (\neg s)$}\\
+ \hline
+ \end{tabular}
+ \end{table}
+
+\item[2.] Apply equivalence and inference rules, particularly disjunctive syllogism (i.e., unit resolution) and the resolution rule, to prove that $q \rightarrow r$.
+
+ \answer
+ \begin{table}[htb]
+ \centering
+ \tabcolsep=0.15cm
+ \begin{tabular}{l l | l}
+ \hline
+ 1. & $(\neg p \vee \neg t \vee r \vee s) \wedge (\neg q \vee u) \wedge (\neg q \vee t) \wedge (\neg u \vee p) \wedge (\neg s)$ & Premise \\
+ 2. & $((\neg p \vee \neg t \vee r \vee s) \wedge (\neg s)) \wedge (\neg q \vee u) \wedge (\neg q \vee t) \wedge (\neg u \vee p)$ & Associative \\
+ 3. & $(\neg p \vee \neg t \vee r) \wedge (\neg q \vee u) \wedge (\neg q \vee t) \wedge (\neg u \vee p)$ & Resolution \\
+ 4. & $(\neg u \vee p) \wedge (\neg p \vee \neg t \vee r) \wedge (\neg q \vee u) \wedge (\neg q \vee t)$ & Associative \\
+ 5. & $(\neg u) \wedge (\neg t \vee r) \wedge (\neg q \vee u) \wedge (\neg q \vee t)$ & Resolution \\
+ 6. & $(\neg q \vee t) \wedge (\neg t \vee r) \wedge (\neg q \vee u) \wedge (\neg u)$ & Associative \\
+ 7. & $(\neg q) \wedge (r) \wedge (\neg q)$ & Resolution \\
+ 8. & $(\neg q) \wedge (\neg q) \wedge (r)$ & Associative \\
+ 9. & $(\neg q) \wedge (r)$ & Idempotent \\
+ 10. & $(q \rightarrow r)$ & Implication \\
+ \hline
+ \end{tabular}
+ \end{table}
+
+\end{enumerate}
+
+\newpage
+
+\problem (\#14 S1.6 Pg79) For each of these arguments, explain which rules of inference are used for each step.
+
+\begin{enumerate}
+\item[a)]``Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.''
+ \begin{enumerate}
+ \item[]\answer Modus Ponens gives ``Linda, a student, has gotten a speeding ticket,'' Existential Generalization gives ``Someone has gotten a speeding ticket,'' Existential Instantiation gives ``Someone in this class has gotten a speeding ticket.''
+ \end{enumerate}
+
+\item[b)]``Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year.''
+
+ \begin{enumerate}
+ \item[]\answer Modus Ponens gives ``Each of the five roommates can take a course in algorithms,'' Universal Generalization gives ``All five roommates can take a course in algorithms.'' Addition gives ``All five roommates can take a course in algorithms next year.''
+ \end{enumerate}
+
+\item[c)]``All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.''
+
+ \begin{enumerate}
+ \item[]\answer Universal Instantiation gives, ``A movie about coal miners produced by John Sayles is wonderful,'' Modus Ponens gives, ``A movie about coal miners is wonderful,'' Existential Generalization gives ``There is a movie about coal miners that is wonderful.''
+ \end{enumerate}
+
+\item[d)]``There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.''
+
+ \begin{enumerate}
+ \item[]\answer Universal Instantiation gives ``Someone in this class who goes to France visits the Louvre,'' Modus Ponens gives ``There is someone in this class who has visited the Louvre.''
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem (\#16 S1.6 Pg79) For each of these arguments determine whether the argument is correct or incorrect and explain why.
+
+\begin{enumerate}
+\item[b)]A convertible car is fun to drive. Isaac’s car is not a convertible. Therefore, Isaac’s car is not fun to drive.
+
+ \begin{enumerate}
+ \item[]\answer This is a fallacy since Isaac's car may still be fun to drive even if it is not a convertible. It affirms a conclusion.
+ \end{enumerate}
+
+\item[c)]Quincy likes all action movies. Quincy likes the movie Eight Men Out. Therefore, Eight Men Out is an action movie.
+
+ \begin{enumerate}
+ \item[]\answer This is a fallacy, Quincy may like other movies as well. It denies the hypothesis.
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem (\#24 S1.6 Pg80) Identify the error or errors in this argument that supposedly shows that if $\forall x (P(x) \vee Q(x))$ is true then $\forall x P(x) \vee \forall x Q(x)$ is true.\vspace{0.5cm}
+
+\centering
+\begin{tabular}{l l l}
+ 1. & $\forall x (P(x) \vee Q(x))$ & Premise\\
+ 2. & $P(c) \vee Q(c)$ & Universal instantiation form (1)\\
+ 3. & $P(c)$ & Simplification form (2)\\
+ 4. & $\forall x P(x)$ & Universal generalization form (3)\\
+ 5. & $Q(c)$ & Simplification form (2)\\
+ 6. & $\forall x Q(x)$ & Universal generalization form (5)\\
+ 7. & $\forall x (P(x) \vee \forall x Q(x))$ & Conjunction form (4) and (6)\\
+\end{tabular}
+\vspace{0.5cm}
+
+\raggedright
+
+\ansindent\answer Step 5: cannot assume that P(c) simplifies to Q(c). They are different quantifiers.
+
+\problem (\#28 S1.6 Pg80) Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$ and\\$\forall x ((\neg P(x) \vee Q(x)) \rightarrow R(x))$ are true, then $\forall x (\neg R(x) \rightarrow P(x))$ is also true where the domains of all quantifiers are the same.\vspace{0.5cm}
+
+\ansindent\answer
+\begin{center}
+\begin{tabular}{ l l | l}
+ \hline
+ 1. & $\forall x (P(x) \vee Q(x))$ & Assumption\\
+ 2. & $\forall x ((\neg P(x) \vee Q(x)) \rightarrow R(x))$ & Assumption\\
+ 3. & $P(a) \vee Q(a)$ & Universal Instantiation (1)\\
+ 4. & $(\neg P(a) \vee Q(a)) \rightarrow R(a)$ & Universal Instantiation (2)\\
+ 5. & $\neg (\neg P(a) \vee Q(a)) \vee R(a)$ & Implication Law (4)\\
+ 6. & $(P(a) \wedge \neg Q(a)) \vee R(a)$ & DeMorgan's Law (5)\\
+ 7. & $P(a) \wedge P(a) \vee R(a)$ & Resolution (3)(6) \\
+ 8. & $R(a) \vee P(a)$ & Idempotent \& Commutative Law (7)\\
+ 9. & $\neg R(a) \rightarrow P(a)$ & Implication Law (8)\\
+ 10.& $\forall x (\neg R(x) \rightarrow P(x))$ & Universal Generalization (9)\\
+ \hline
+\end{tabular}
+\end{center}
+
+\end{document}
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--- /dev/null
+++ b/hw05/HW05DanielShchur.tex
@@ -0,0 +1,310 @@
+\documentclass[12pt,letterpaper]{article}
+
+\usepackage{bm}
+\usepackage[letterpaper, margin=.75in]{geometry}
+\usepackage{amssymb}
+\usepackage{multicol}
+
+\newcommand{\problem}{\vspace{.5cm}\textbf{Problem:~}}
+\newcommand{\problemEnum[1]}{\vspace{.5cm}\textbf{Problem {#1}:~}}
+\newcommand{\answer}{\emph{Answer:~~}}
+\newcommand{\ansindent}{\indent\hspace{0.5cm}}
+\newcommand{\ansend}{\vspace{0.25cm}\\}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0cm}
+\setlength{\intextsep}{.25cm}
+
+\title{CSCE 235H Homework 05}
+\author{Daniel Shchur}
+
+\begin{document}
+
+\maketitle
+
+\problemEnum[A] Using contraposition, prove that
+\[\forall x,y \in \mathbb{R}, x+y \geq 2 \textrm{, then } x \geq 1 \textrm{ or } y \geq 1.\]
+
+\begin{table}[htb]
+\centering
+\begin{tabular}{l l | l}
+ 1. & $x \geq 1$ or $y \geq 1$ & Premise \\
+ 2. & $x < 1$ or $y < 1$ & DeMorgan's law (1)\\
+ 3. & $x + y < 2$ & Adding the two parts (2)\\
+\end{tabular}
+\end{table}
+
+\problemEnum[B] Write the following sentences in First Order Logic, first, declaring the UoD and the
+predicates, their arguments and meaning in English:
+
+\begin{enumerate}
+\item For every mall, there is some Santa who is at the mall.
+ \begin{enumerate}
+ \item[] \answer Where all malls and all Santas are part of the UoD\\
+ Let $S(x)$ denote ``$x$ is a Santa,'' $M(x)$ denote ``$x$ is a mall,'' and \\$A(x,y)$ denote ``$x$ is at $y$''\\
+ Therefore: $\exists x \forall y ((S(x) \wedge M(y)) \rightarrow A(x,y)$
+
+ \end{enumerate}
+
+\item Every child who visits anywhere talks with every Santa who is at the place visited. [Don’t make a predicate for ``the place visited;'' it should just be a variable.]
+ \begin{enumerate}
+ \item[] \answer Where all children, all Santas, and all places are part of the UoD\\
+ Let $S(x)$ denote ``$x$ is a Santa,'' $C(x)$ denote ``$x$ is a child,'' $P(x)$ denote ``$x$ is a place,'' $T(x,y)$ denote ``$x$ talks to $y$,'' and $A(x,y)$ denote ``$x$ is at $y$.''\\
+ Therefore: $\exists x \forall y,z ((S(x) \wedge C(y) \wedge P(z)) \rightarrow ((A(y,z) \wedge A(x,z)) \rightarrow T(y,x))$
+ \end{enumerate}
+
+\item Every child who is a city child visits some mall.
+ \begin{enumerate}
+ \item[] \answer Where all malls and all children are part of the UoD\\
+ Let $K(x)$ denote ``is a child,'' $M(x)$ denote ``is a mall,'' $C(x)$ denote ``is from the city,'' and $A(x,y)$ denote ``$x$ visits $y$.''\\ Therefore: $\exists x \forall y ((K(y) \wedge C(y) \wedge M(x)) \rightarrow A(y,x)$
+ \end{enumerate}
+
+ \newpage
+
+\item Every child who is good or who talks with some Santa gets some toy.
+ \begin{enumerate}
+ \item[] \answer Where all children, all Santas, and all toys are part of the UoD\\
+ Let $S(x)$ denote ``$x$ is a Santa,'' $C(x)$ denote ``$x$ is a child,'' $T(x)$ denote ``$x$ is a toy,'' $G(x)$ denote ``$x$ is good,'' $P(x,y)$ denote ``$x$ talks to $y$,'' and \\ $Q(x,y)$ denote ``$x$ receives $y$.''\\
+ Therefore: $\exists x,z \forall y ((C(y) \wedge S(y) \wedge T(z)) \rightarrow ((P(y,x) \vee G(y)) \rightarrow Q(y,z))$
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem (\#30 S1.7 Pg91) Show that these three statements are equivalent, where $a$ and $b$ are real numbers: $(i)$ $a$ is less than $b$, $(ii)$ the average of $a$ and $b$ is greater than $a$, and $(iii)$ the average of $a$ and $b$ is less than $b$.
+
+\begin{enumerate}
+\item[] To prove that all 3 statments are equivalent, we must prove that $(i) \leftrightarrow (ii) \wedge (i) \leftrightarrow (iii)$.\\
+
+ To start, assume $(i)$: Where $a < b$\\
+ Start by adding a to both sides: $2a < a + b$\\
+ Then use division to get: $a < \frac{a+b}{2}$, which happens to be $(ii)$ and true.\\
+
+ Then assume $(i)$ again: Where $a < b$\\
+ This time, add b to both sides: $a+b < 2b$\\
+ Then use division to get: $\frac{a+b}{2} < b$, which happens to be $(iii)$ and true.\\
+
+ Therefore: $(i) \leftrightarrow (ii) \wedge (i) \leftrightarrow (iii)$ and $ (ii) \leftrightarrow (iii)$ through hypothetical syllogism.
+\end{enumerate}
+
+\problem (\#4 S1.8 Pg108) Use a proof by cases to show that min($a$, min($b$, $c$)) = min(min($a$, $b$), $c$) whenever $a$, $b$, and $c$ are real numbers.
+
+\begin{enumerate}
+\item[] Case 1: Assume $a$ = min($a$, min($b$, $c$)), in that case, $a <$ min($b$, $c$) which implies $a < b \wedge a < c$ therefore, in a general sense showing that in min(min($a$, $b$), $c$), $a < b \wedge a < c$ which is equivalent to the previous statement.\\
+
+ Case 2: Assume $b$ = min($a$, min($b$, $c$)), that case, $b < c \wedge b < a$ therefore in a general sense showing that in min(min($a$, $b$), $c$), $b < a \wedge b < c$, which is equivalent to the previous statement.\\
+
+ Case 3: Assume $c$ = min($a$, min($b$, $c$)), that case, $c < b \wedge c < a$ therefore in a general sense showing that in min(min($a$, $b$), $c$), $c <$ min($a$, $b$) which implies $c < a \wedge c < b$, which is equivalent to the previous statement.
+\end{enumerate}
+
+\problem (\#21 S1.8 Pg108) Prove that given a real number $x$ there exist unique numbers $n$ and $\epsilon$ such that $x = n - \epsilon $, $n$ is an integer, and $0 \leq \epsilon < 1$.
+
+\begin{enumerate}
+\item[] Case 1: If $x$ is an integer, then let $x = n$ and $\epsilon = 0$, therefore, $x = n - \epsilon$ holds true.\\
+
+ Case 2: If $x$ is not an integer, let $n$ be the nearest integer rounded up such that $\epsilon = n - x$, therefore $x = n - \epsilon$ holds true.
+\end{enumerate}
+
+\newpage
+
+\problem (\#36 S1.8 Pg109) Prove that between every rational number and every irrational number there is an irrational number.
+
+\begin{enumerate}
+\item[] Assume $(a,b) \in \mathbb{Z}$, such that $\frac{a}{b}$ is rational when both $a$ and $b$ are divisible by 2.
+
+ Then let $\frac{\sqrt{2} + a}{b}$ be irrational as $\sqrt{2}$ is irrational.
+
+ And let $\frac{\frac{1}{2}\sqrt{2} + a}{b}$ be irrational as well.
+
+ Where $\frac{a}{b} < \frac{\sqrt{2} + a}{b}$ and $\frac{\frac{1}{2}\sqrt{2} + a}{b} < \frac{\sqrt{2} + a}{b}$ so that $\frac{a}{b} < \frac{\frac{1}{2}\sqrt{2} + a}{b} < \frac{\sqrt{2} + a}{b}$.
+
+ Therefore, there is an irrational number between every rational and irrational number.
+\end{enumerate}
+
+\problem (\#2 S2.1 Pg125) Use set builder notation to give a description of each of these sets.
+
+\begin{enumerate}
+\item[(a)] \{0, 3, 6, 9, 12\}
+ \begin{enumerate}
+ \item[] \answer $ S = \{x | x \in \mathbb{Z}, 0 \leq x \leq 12 \wedge (x=3k), k \in \mathbb{Z}\}$
+ \end{enumerate}
+
+\item[(b)] \{-3, -2, -1, 0, 1, 2, 3\}
+ \begin{enumerate}
+ \item[] \answer $S = \{x| x \in \mathbb{Z}, -3 \leq x \leq 3\}$
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem (\#6 S2.1 Pg125) Suppose that $A = \{2, 4, 6\}$, $B = \{2, 6\}$, $C = \{4, 6\}$, and $D = \{4, 6, 8\}$. Determine which of these sets are subsets of which other of these sets.
+
+\begin{enumerate}
+\item[] \answer $B \subset A$, $C \subset (A \wedge D)$
+\end{enumerate}
+
+\problem (\#10 S2.1 Pg125) Determine whether these statements are true or false.
+
+\begin{multicols}{2}
+\begin{enumerate}
+\item[(a)] $\emptyset \in \{\emptyset\}$
+ \begin{enumerate}
+ \item[] \answer True
+ \end{enumerate}
+
+\item[(b)] $\emptyset \in \{\emptyset, \{\emptyset\}\}$
+ \begin{enumerate}
+ \item[] \answer True
+ \end{enumerate}
+
+\item[(c)] $\{\emptyset\} \in \{\emptyset\}$
+ \begin{enumerate}
+ \item[] \answer False
+ \end{enumerate}
+
+\item[(d)] $\{\emptyset\} \in \{\{\emptyset\}\}$
+ \begin{enumerate}
+ \item[] \answer True
+ \end{enumerate}
+
+\item[(e)] $\{\emptyset\} \subset \{\emptyset , \{\emptyset\}\}$
+ \begin{enumerate}
+ \item[] \answer True
+ \end{enumerate}
+
+\item[(f)]$\{\{\emptyset\}\} \subset \{\emptyset , \{\emptyset\}\}$
+ \begin{enumerate}
+ \item[] \answer False
+ \end{enumerate}
+
+\item[(g)] $\{\{\emptyset\}\} \subset \{\{\emptyset\}, \{\emptyset\}\}$
+ \begin{enumerate}
+ \item[] \answer False
+ \end{enumerate}
+
+\end{enumerate}
+\end{multicols}
+
+\newpage
+
+\problem (\#20 S2.1 Pg126) What is the cardinality of each of these sets?
+
+\begin{multicols}{2}
+\begin{enumerate}
+\item[(a)] $\emptyset$
+ \begin{enumerate}
+ \item[] \answer 0
+ \end{enumerate}
+
+\item[(b)] $\{\emptyset\}$
+ \begin{enumerate}
+ \item[] \answer 1
+ \end{enumerate}
+
+\item[(c)] $\{\emptyset, \{\emptyset\}\}$
+ \begin{enumerate}
+ \item[] \answer 2
+ \end{enumerate}
+
+\item[(d)] $\{\emptyset, \{\emptyset\}, \{\emptyset , \{\emptyset\}\}\}$
+ \begin{enumerate}
+ \item[] \answer 3
+ \end{enumerate}
+\end{enumerate}
+\end{multicols}
+
+\problem (\#24 S2.1 Pg126) Determine whether each of these sets is the power set of a set, where $a$ and $b$ are distinct elements.
+
+\begin{multicols}{2}
+\begin{enumerate}
+\item[(a)] $\emptyset$
+ \begin{enumerate}
+ \item[] \answer No
+ \end{enumerate}
+
+\item[(b)] $\{\emptyset , \{a\}\}$
+ \begin{enumerate}
+ \item[] \answer Yes
+ \end{enumerate}
+
+\item[(c)] $\{\emptyset , \{a\}, \{\emptyset , a\}\}$
+ \begin{enumerate}
+ \item[] \answer No
+ \end{enumerate}
+
+\item[(d)] $ \{\emptyset , \{a\}, \{b\}, \{a,b\}\}$
+ \begin{enumerate}
+ \item[] \answer Yes
+ \end{enumerate}
+\end{enumerate}
+\end{multicols}
+
+\problem (\#26 S2.1 Pg126) Show that if $A \subseteq C$ and $B \subseteq D$, then $A \times B \subseteq C \times D$
+
+\begin{enumerate}
+\item[] \answer Take any $(a,b)$ where $a \in A$ and $b \in B$ where $A \times B = \{ab\}$. As $A \subseteq C$, then $a \in C$ and as $B \subseteq D$, then $b \in C$. Therefore, $C \times D = \{ab\} \equiv A \times B$
+\end{enumerate}
+
+\problem (\#32 S2.1 Pg126) Let $A = \{a,b,c\}$, $B = \{x,y\}$, and $C = \{0,1\}$.
+
+\begin{enumerate}
+\item[(a)] $A \times B \times C$
+ \begin{enumerate}
+ \item[] \answer $\{(a,x,0), (a,x,1), (a,y,0), (a,y,1), (b,x,0), (b,x,1), (b,y,0), (b,y,1), (c,x,0),\\ (c,x,1), (c,y,0), (c,y,1)\}$
+ \end{enumerate}
+
+\item[(c)] $C \times A \times B$
+ \begin{enumerate}
+ \item[] \answer $\{(0,a,x), (0,a,y), (0,b,x), (0,b,y), (0,c,x), (0,c,y), (1,a,x), (1,a,y), (1,b,x),\\ (1,b,y), (1,c,x), (1,c,y)\}$
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem (\#38 S2.1 Pg126) Show that $A \times B \neq B \times A$, when $A$ and $B$ are nonempty, unless $A = B$.
+
+\begin{enumerate}
+\item[] \answer This is the same rule that applies to matrices. Where $(a,b) \in A$ and $(c,d) \in B$. $A \times B = \{(a,c), (a,d), (b,c), (b,d)\}$ and $B \times A = \{(c,a), (c,b), (d,a), (d,b)\}$ which is not the same. Therefore, $A \times B \neq B \times A$.
+\end{enumerate}
+
+\newpage
+
+\problem (\#40 S2.1 Pg126) Explain why $(A \times B) \times (C \times D)$ and $A \times (B \times C) \times D$ are not the same.
+
+\begin{enumerate}
+\item[] \answer The two are not the same since order of operations applies where everything in parenthesis is carried out first, followed by the rest. As was proved in the previous problem, the order by which sets are multiplied matters greatly, so taking $B \times C$ first, as is the case in the second problem, changes the rest of the problem, making it not equal to the first problem.
+\end{enumerate}
+
+\problem (\#42 S2.1 Pg126) Translate each of these quantifications into English and determine its truth value.
+
+\begin{enumerate}
+\item[(a)] $\exists x \in \mathbb{R} (x^3 = -1)$
+ \begin{enumerate}
+ \item[] \answer There is an $x$ that is a real number where $x^3 = -1$. That is true since $(-1)^3 = -1$.
+ \end{enumerate}
+
+\item[(c)] $\forall x \in \mathbb{Z} (x - 1 \in \mathbb{Z})$
+ \begin{enumerate}
+ \item[] \answer For all $x$ that are integers, $x-1$ results in an integer. This is true as an integer minus and integer will always result in an integer.
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem (\#44 S2.1 Pg126) Find the truth set of each of these predicates where the domain is the set of integers.
+
+\begin{enumerate}
+\item[(a)] $P(x): x^3 \geq 1$
+ \begin{enumerate}
+ \item[] \answer Truth set: $S = \{x | x \in \mathbb{Z}, x \geq 1\}$
+ \end{enumerate}
+
+\item[(b)] $Q(x): x^2 = x$
+ \begin{enumerate}
+ \item[] \answer Truth set: $S = \emptyset$
+ \end{enumerate}
+
+\item[(c)] $R(x): x < x^2$
+ \begin{enumerate}
+ \item[] \answer Truth set: $S = \{x | x \in \mathbb{Z}\}$
+ \end{enumerate}
+
+\end{enumerate}
+
+\end{document}
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+\documentclass[12pt,letterpaper]{article}
+
+\usepackage{bm}
+\usepackage[letterpaper, margin=.75in]{geometry}
+\usepackage{amssymb}
+\usepackage{multicol}
+\usepackage{graphicx}
+
+\newcommand{\problem[2]}{\vspace{.25cm}\textbf{Problem {#1}}\footnotesize{#2}\textbf{:~}\normalsize}
+\newcommand{\answer}{\emph{Answer:~~}}
+\newcommand{\ansindent}{\indent\hspace{0.5cm}}
+\newcommand{\ansend}{\vspace{0.25cm}\\}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0cm}
+\setlength{\intextsep}{.25cm}
+
+\title{CSCE 235H Homework 06}
+\author{Daniel Shchur}
+
+\begin{document}
+
+\maketitle
+
+\problem[\#4]{ (S2.2 Pg136)} Let A = \{$a$, $b$, $c$, $d$, $e$\} and $B$ = \{$a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$\}. Find
+
+\begin{multicols}{2}
+\begin{enumerate}
+\item[(a)] $A \cup B$
+ \begin{enumerate}
+ \item[] \answer \{$a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$\}
+ \end{enumerate}
+
+\item[(b)] $A \cap B$
+ \begin{enumerate}
+ \item[] \answer \{$a$, $b$, $c$, $d$, $e$\}
+ \end{enumerate}
+
+\item[(c)] $A \-- B$
+ \begin{enumerate}
+ \item[] \answer $\emptyset$
+ \end{enumerate}
+
+\item[(d)] $B \-- A$
+ \begin{enumerate}
+ \item[] \answer \{$f$, $g$, $h$\}
+ \end{enumerate}
+\end{enumerate}
+\end{multicols}
+
+
+\problem[\#14]{ (S2.2 Pg136)} Find the sets $A$ and $B$ if $A \-- B$ = \{1, 5, 7, 8\}, $B \-- A$ = \{2, 10\}, and $A \cap B$ = \{3, 6, 9\}.
+
+\begin{enumerate}
+\item[] If $A \-- B$ = \{1, 5, 7, 8\}, then \{1, 5, 7, 8\} $\in A$ and $\notin B$\ansend
+ Also, if $B \-- A$ = \{2, 10\}, then \{2, 10\} $\in B \wedge$ and $\notin A$\ansend
+ Finally, if $A \cap B$ = \{3, 6, 9\}, then \{3, 6, 9\} $\in A$ and $\in B$\ansend
+ Therefore:\\
+ \ansindent $A$ = \{1, 3, 5, 6, 7, 8, 9\}\\
+ \ansindent $B$ = \{2, 3, 6, 9, 10\}
+\end{enumerate}
+
+\problem[\#16]{ (S2.2 Pg136)} Let $A$ and $B$ be sets. Show that
+
+\begin{enumerate}
+\item[(a)] $(A \cap B) \subseteq A$
+ \begin{enumerate}
+ \item[] Suppose $x \in A \cap B$, then by definition of intersection, $x \in A \wedge x \in B$.\\
+ By definition of a subset, $x \in A$.\\
+ Therefore $(A \cap B) \subseteq A$.
+ \end{enumerate}
+
+\item[(c)] $A \-- B \subseteq A$
+ \begin{enumerate}
+ \item[] Suppose $x \in A \-- B$, then by definition of difference, $x \in A \wedge x \notin B$.\\
+ By definition of a subset, $x \in A$.\\
+ Therefore: $A \-- B \subseteq A$.
+ \end{enumerate}
+\newpage
+\item[(e)] $A \cup (B \-- A) = A \cup B$
+ \begin{enumerate}
+ \item[] Suppose $x \in B \-- A$, then by definition of difference, $x \in B \wedge x \notin A$.\\
+ By definition of union, $x \in A \vee x \in B$.\\
+ As $x \in B$, $x \in A \vee x \in B$ is true.\\
+ Therefore $A \cup (B \-- A) = A \cup B$.
+ \end{enumerate}
+\end{enumerate}
+
+\problem[\#22]{ (S2.2 Pg136)} Prove the second associative law from Table 1 by showing that if $A$, $B$, and $C$ are sets, then $A \cap (B \cap C) = (A \cap B) \cap C$.
+
+\begin{enumerate}
+ \item[] Suppose $x \in B \cap C$, then by definition of intersection, $x \in B \wedge x \in C$.\\
+ By definition of intersection, $x \in A \wedge x \in B \cap C$.\\
+ Since $x \in A, B, C$, any intersection of the sets, by definition of intersection, will be possible.\\
+ Therefore $A \cap (B \cap C) = (A \cap B) \cap C$.
+ \end{enumerate}
+
+\problem[\#28]{ (S2.2 Pg136)} Draw the Venn diagrams for each of these combinations of the sets $A$, $B$, $C$, and $D$.
+\begin{multicols}{2}
+\begin{enumerate}
+\item[(a)] $(A \cap B) \cup (C \cap D)$
+ \begin{enumerate}
+ \item[]\includegraphics[scale=0.16]{p28A.png}
+ \end{enumerate}
+
+\begin{minipage}{\textwidth}
+\item[(b)] $\overline{A} \cup \overline{B} \cup \overline{C} \cup \overline{D}$
+ \begin{enumerate}
+ \item[]\includegraphics[scale=0.16]{p28B.png}
+ \end{enumerate}
+\end{minipage}
+
+\item[(c)] $A \-- (B \cap C \cap D)$
+ \begin{enumerate}
+ \item[]\includegraphics[scale=0.16]{p28C.png}
+ \end{enumerate}
+
+\end{enumerate}
+\end{multicols}
+
+\newpage
+
+\problem[A]{}\vspace{0.25cm}
+
+Following the steps below:
+\begin{enumerate}
+\item State the propositions and their meaning in English.
+\item Write each statement in propositional logic and number each statement.
+\item Then, use the rules of inference to derive the conclusion from the premises clearly documenting each step.
+\end{enumerate}
+Determine whether or not the following argument is valid:
+\begin{enumerate}
+\item She is a Math Major or a Computer Science Major.
+\item If she does not know discrete math, she is not a Math Major.
+\item If she knows discrete math, she is smart.
+\item She is not a Computer Science Major.
+\item Therefore, she is smart.
+\end{enumerate}
+
+\answer
+
+\begin{center}
+ \begin{tabular}{l l | l}
+ & She is a math major. & $m$\\
+ & She is a Comp Sci major. & $c$\\
+ & She knows discrete math. & $d$\\
+ & She is smart. & $s$\\
+ \hline
+ 1. & She is a Math Major or a Computer Science Major. & $m \vee c$ \\
+ 2. & If she does not know discrete math, she is not a Math Major. & $\neg d \rightarrow \neg m$ \\
+ 3. & If she knows discrete math, she is smart. & $d \rightarrow s$\\
+ 4. & She is not a Computer Science Major. & $\neg c$\\
+ \hline
+ 5. & Therefore, she is smart. & $s$\\
+ \end{tabular}
+\end{center}
+
+\vspace{0.25cm}
+
+\begin{center}
+ \begin{tabular}{l l | l}
+ 1. & $m \vee c$ & Premise \\
+ 2. & $\neg d \rightarrow \neg m$ & Premise \\
+ 3. & $d \rightarrow s$ & Premise\\
+ 4. & $\neg c$ & Premise\\
+ \hline
+ 5. & $m$ & Unit Resolution (1)(4)\\
+ 6. & $m \rightarrow d$ & Law of Contrapositive (2)\\
+ 7. & $d$ & Modus Ponens (6)(5)\\
+ 8. & $s$ & Modus Ponens (7)(3)\\
+ \hline
+ \multicolumn{3}{c}{QED}
+ \end{tabular}
+\end{center}
+
+\newpage
+
+\problem[B]{} Suppose you wish to prove a theorem of the form ``if $p$ then $q$.''
+\begin{enumerate}
+\item If you give a direct proof, what do you assume and what do you prove?
+ \begin{enumerate}
+ \item[] You prove directly the original premise to its conclusion, if $p \rightarrow q$ is the premise, prove that it is true.
+ \end{enumerate}
+\item If you give a proof by contraposition, what do you assume and what do you prove?
+ \begin{enumerate}
+ \item[] You prove the contrapositive of the premise where if $p \rightarrow q$ is the premise, you assume $\neg q \rightarrow \neg p$.
+ \end{enumerate}
+\item If you give a proof by contradiction, what do you assume and what do you prove?
+ \begin{enumerate}
+ \item[] You assume the opposite of the premise, where, if $p \rightarrow q$ is the premise, you assume $\neg p \rightarrow q$, and prove that such a premise would not be possible, proving the original true.
+ \end{enumerate}
+\end{enumerate}
+
+\problem[C]{} Give a proof by contradiction of the following:
+\begin{center}
+ ``If $n$ is an odd integer, then $n^2$ is odd.''
+\end{center}
+
+\begin{enumerate}
+\item Assume $n$ is an even integer, but $n^2$ is odd.
+\item $n = 2k | k \in \mathbb{Z}$
+\item $(2k)^2 = n^2 \equiv 4k^2 = n^2$
+\item Let $m = 2k^2$ so that $2m = n^2$
+\item By definition, n is even; therefore, the conclusion is false.
+\end{enumerate}
+
+\problem[D]{} Prove that the following is true for all positive integers $n$:
+\begin{center}
+ $n$ is even if and only if $3n^2 + 8$ is even.
+\end{center}
+
+\begin{center}
+\begin{tabular}{l l | l}
+ \hline
+ 1. & $n$ is even & Premise\\
+ 2. & $\leftrightarrow n = 2k | k \in \mathbb{Z}$ & Definition of even integers\\
+ 3. & $\leftrightarrow n + 1 = 2k + 1$ & Algebra\\
+ 4. & $\leftrightarrow (n + 1)(n - 1) = (2k + 1)(2k - 1)$ & Algebra\\
+ 5. & $\leftrightarrow n^2 - 1 = 4k^2 - 1$ & Algebra\\
+ 6. & $\leftrightarrow 3n^2 - 3 = 12k^2 - 3$ & Algebra\\
+ 7. & $\leftrightarrow 3n^2 + 8 = 12k^2 + 8$ & Algebra\\
+ 8. & $\leftrightarrow 3n^2 + 8 = 2(6k^2 + 4)$ & Factoring\\
+ \hline
+ \multicolumn{3}{c}{As the right side is a positive even integer and has stayed throughout,}\\
+ \multicolumn{3}{c}{the conclusion must be true.}\\
+\end{tabular}
+\end{center}
+
+\newpage
+
+\problem[E]{} Give a proof by contraposition of the following:
+\begin{center}
+ If $3n + 5$ is even, then $n$ is odd.
+\end{center}
+
+\begin{enumerate}
+\item Assume: If $n$ is even, then $3n + 5$ is odd.
+\item Even: $n = 2k | k \in \mathbb{Z}$, Odd: $m = 2k + 1 | k \in \mathbb{Z}$
+\item $3(2k) + 5 \equiv 6k + 5 \equiv 4(2k + 1) + 1 \equiv 4m + 1$
+\item Conclusion is odd; therefore, by law of contrapositive, the statement is true.
+\end{enumerate}
+
+\problem[F]{} Prove that the following three statements about positive integers $n$ are equivalent:
+\begin{enumerate}
+\item $n$ is even
+ \begin{enumerate}
+ \item[]
+ \begin{tabular}{l l | l}
+ 1. & $n = 2k | k \in \mathbb{Z}$ & Definition of even integer\\
+ \end{tabular}
+ \end{enumerate}
+\item $n^3 + 1$ is odd
+ \begin{enumerate}
+ \item[]
+ \begin{tabular}{l l | l}
+ \hline
+ 1. & $n = 2k | k \in \mathbb{Z}$ & Definition of even integer\\
+ 2. & $\leftrightarrow n^3 = 8k^3$ & Multiplication\\
+ 3. & $\leftrightarrow n^3 + 1 = 8k^3 + 1$ & Algebra\\
+ 4. & $\leftrightarrow n^3 + 1 = 8k^3 + 1$ & Even integer + 1 is odd by definition\\
+ \hline
+ \multicolumn{3}{c}{Therefore: If $n^3 + 1$ is odd, then $n$ is even.}\\
+ \end{tabular}
+ \end{enumerate}
+\item $n^2 - 1$ is odd
+ \begin{enumerate}
+ \item[]
+ \begin{tabular}{l l | l}
+ \hline
+ 1. & $n = 2k | k \in \mathbb{Z}$ & Definition of even integer\\
+ 2. & $\leftrightarrow n^2 = 4k^2$ & Multiplication\\
+ 3. & $\leftrightarrow n^2 - 1 = 4k^2 - 1$ & Algebra\\
+ 4. & $\leftrightarrow n^2 + 1 = 4k^2 - 1$ & Even integer $-$ 1 is odd by definition\\
+ \hline
+ \multicolumn{3}{c}{Therefore: If $n^2 - 1$ is odd, then $n$ is even.}\\
+ \end{tabular}
+ \end{enumerate}
+\end{enumerate}
+
+\begin{center}
+ As all 3 statements conclude to $n$ is even, all must be equivalent.
+\end{center}
+
+\problem[G]{} Using a proof by cases, prove that:
+\begin{center}
+ The equation $2x^2 + y^2 = 14$ has no positive integer solutions.
+\end{center}
+
+\begin{multicols}{2}
+\begin{enumerate}
+\item $x = 1$, $y = 1$: $2(1)^2 + (1)^2 = 3 \neq 14$
+\item $x = 1$, $y = 2$: $2(1)^2 + (2)^2 = 6 \neq 14$
+\item $x = 1$, $y = 3$: $2(1)^2 + (3)^2 = 11 \neq 14$
+\item $x = 2$, $y = 1$: $2(2)^2 + (1)^2 = 9 \neq 14$
+\item $x = 2$, $y = 2$: $2(2)^2 + (2)^2 = 12 \neq 14$
+\item $x = 2$, $y = 3$: $2(2)^2 + (3)^2 = 17 \neq 14$
+\end{enumerate}
+
+Since positive integers include all numbers above 0, the smallest both $x$ and $y$ can be is 1. The largest $x$ can be is 2 as any larger, and the sum would be greater than 14. With $y$, it can't be larger than 3. After going through all the cases, no combination of $x$ and $y$ where they are both positive integers resulted in 14. Therefore: The equation $2x^2 + y^2 = 14$ has no positive integer solutions.
+\end{multicols}
+
+\end{document}
\ No newline at end of file
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+\documentclass[12pt,letterpaper]{article}
+
+\usepackage{bm}
+\usepackage[letterpaper, margin=.75in]{geometry}
+\usepackage{amssymb}
+\usepackage{multicol}
+
+\newcommand{\problem[2]}{\vspace{.25cm}\textbf{Problem {#1}}\footnotesize{#2}\textbf{:~}\normalsize}
+\newcommand{\answer}{\emph{Answer:~~}}
+\newcommand{\ansindent}{\indent\hspace{0.5cm}}
+\newcommand{\ansend}{\vspace{0.25cm}\\}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0cm}
+\setlength{\intextsep}{.25cm}
+
+\title{CSCE 235H Homework 07}
+\author{Daniel Shchur}
+
+\begin{document}
+
+\maketitle
+
+\problem[\#2]{ (S2.3 Pg152)} Determine whether $f$ is a function from $\mathbb{Z}$ to $\mathbb{R}$ if:
+\begin{enumerate}
+\item[a)] $f(n) = \pm n$
+ \begin{enumerate}
+ \item[] \answer No
+ \end{enumerate}
+
+\item[b)] $f(n) = \sqrt{n^2 + 1}$
+ \begin{enumerate}
+ \item[] \answer Yes
+ \end{enumerate}
+
+\item[c)] $f(n) = \frac{1}{(n^2 \-- 4)}$
+ \begin{enumerate}
+ \item[] \answer No
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem[\#20]{ (S2.3 Pg153)} Give an example of a function from $\mathbb{N}$ to $\mathbb{N}$ that is
+\begin{enumerate}
+\item[a)] one-to-one but not onto.
+ \begin{enumerate}
+ \item[] \answer $f(x) = x + 1$ where $x \geq 0$
+ \end{enumerate}
+
+\item[b)] onto but not one-to-one.
+ \begin{enumerate}
+ \item[] \answer $f(x) = \lfloor \frac{x}{2} \rfloor$ where $x \geq 0$
+ \end{enumerate}
+
+\item[c)] both onto and one-to-one (but different from the identity function).
+ \begin{enumerate}
+ \item[] \answer Where $ x \geq 0$, let $f(x) = x + 1$ where $x$ is even or 0 and let $f(x) = x - 1$ where $x$ is odd.
+ \end{enumerate}
+
+\item[d)] neither one-to-one nor onto.
+ \begin{enumerate}
+ \item[] \answer $f(x) = \lfloor \frac{x}{2} + 1 \rfloor$ where $x \geq 0$
+ \end{enumerate}
+
+\end{enumerate}
+
+\newpage
+
+\problem[\#22]{ (S2.3 Pg153)} Determine whether each of these functions is a bijection from $\mathbb{R}$ to $\mathbb{R}$.
+
+\begin{enumerate}
+\item[a)] $f(x) = -3x + 4$
+ \begin{enumerate}
+ \item[] \answer Yes
+ \end{enumerate}
+
+\item[c)] $f(x) = \frac{(x + 1)}{(x + 2)}$
+ \begin{enumerate}
+ \item[] \answer No
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem[\#34]{ (S2.3 Pg154)} If $f$ and $f \circ g$ are one-to-one, does it follow that $g$ is one-to-one? Justify your answer.
+\begin{enumerate}
+\item[] \answer Yes, let's assume that $g$ is not one-to-one and thus, we need to show that $f$ is not one-to-one as well:\\
+ 1. We take the negation of the definition of one-to-one functions: $\exists a,b (a \neq b \wedge g(a) = g(b))$\\
+ 2. Therefore, $f(g(a)) = f(g(b))$ and thus, $f \circ g$ is not one-to-one.\\
+ 3. This is contradictory, and thus, shows that $g$ must be one-to-one.
+\end{enumerate}
+
+\problem[\#40]{ (S2.3 Pg154)} Let $f$ be a function from the set $A$ to the set $B$. Let $S$ and $T$ be subsets of $A$. Show that:
+\begin{enumerate}
+\item[a)] $f (S \cup T) = f (S) \cup f (T)$.
+ \begin{enumerate}
+ \item[] \answer We need to show that $f(S \cup T) \subseteq f(S) \cup f(T)$ and $f(S) \cup f(T) \subseteq f(S \cup T)$.\\\\
+ 1. Suppose $b \in f(S \cup T)$. Then $\exists a (f(a) = b\ |\ a \in S \cup T)$\\
+ 2. As $a \in S \cup T$, then $a \in S \vee T \vee (S \wedge T)$\\
+ 3. Therefore $b \in f(S) \vee f(T) \vee (f(S) \wedge f(T)$, so $b \in f(S) \cup f(T)$\\\\
+ 4. Suppose that $b \in f(S) \cup f(T)$. Then $b \in f(S) \vee f(T) \vee (f(S) \wedge f(T))$\\
+ 5. Therefore, $\exists a, (f(a) = b\ |\ a \in S \vee T \vee (S \wedge T))$.\\
+ 6. As $a \in S \cup T$, then $f(a) = b \in f(S \cup T)$.\\\\
+ 7. Since $f(S \cup T) \subseteq f(S) \cup f(T)$ and $f(S) \cup f(T) \subseteq f(S \cup T)$, $f(S \cup T) = f(S) \cup f(T)$.
+ \end{enumerate}
+
+\item[b)] $f (S \cap T) \subseteq f (S) \cap f (T)$.
+ \begin{enumerate}
+ \item[] \answer 1. Suppose $b \in f(S \cap T)$. Then $\exists a (f(a) = b\ |\ a \in (S \cap T))$\\
+ 2. Since $a \in S \cap T$, then $a \in S \wedge T$\\
+ 3. Therefore $b \in f(S) \cap f(T) \rightarrow f(S \cap T) \subseteq f(S) \cap f(T)$
+ \end{enumerate}
+
+\end{enumerate}
+ \newpage
+\problem[A]{ (\#10 S2.3)} let $f:\{a,b,c,d\} \rightarrow \{a,b,c,d\}$, for each of function $f$ defined below:
+
+\begin{enumerate}
+\item $f(a) = b$, $f(b) = a$, $f(c) = c$, $f(d) = d$.
+\item $f(a) = b$, $f(b) = b$, $f(c) = d$, $f(d) = c$.
+\item $f(a) = d$, $f(b) = a$, $f(c) = c$, $f(d) = d$.
+\end{enumerate}
+
+Determine whether each $f$ is
+
+\begin{enumerate}
+\item[a.] one-to-one (injective)
+ \begin{enumerate}
+ \item[1.] Yes
+ \item[2.] No
+ \item[3.] No
+ \end{enumerate}
+
+\item[b.] onto (surjective)
+ \begin{enumerate}
+ \item[1.] Yes
+ \item[2.] No
+ \item[3.] No
+ \end{enumerate}
+
+\item[c.] one-to-one correspondence (bijective)
+ \begin{enumerate}
+ \item[1.] Yes
+ \item[2.] No
+ \item[3.] No
+ \end{enumerate}
+
+\end{enumerate}
+
+\problem[B]{ (\#12 S2.3)} Consider the following functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$
+
+\begin{itemize}
+\item $f(n) = n \-- 1$
+ \begin{enumerate}
+ \item[] It is injective, surjective, bijective, and $f^{-1}(n) = m + 1$
+ \end{enumerate}
+
+\item $f(n) = n^2 + 1$
+ \begin{enumerate}
+ \item[] It is not injective nor surjective, and therefore, neither bijective or invertible.
+ \end{enumerate}
+
+\item $f(n) = n^3$
+ \begin{enumerate}
+ \item[] It is injective, but not surjective, and therefore, neither bijective or invertible.
+ \end{enumerate}
+
+\item $f(n) = \lceil\frac{n}{2}\rceil$
+ \begin{enumerate}
+ \item[] It is not injective, but it is surjective, and therefore, neither bijective or invertible.
+ \end{enumerate}
+
+\end{itemize}
+
+For \emph{each} of the above functions, answer each of the following questions:
+
+\newpage
+
+\begin{enumerate}
+\item Determine whether each $f$ is one-to-one (injective).
+\item Determine whether each $f$ is onto (surjective).
+\item Determine whether each $f$ is one-to-one correspondence (bijective).
+\item Determine whether each $f$ is invertible. If so, give $f^{-1}$.
+\end{enumerate}
+
+\problem[C]{} For functions $f(x) = x^2 + x$ and $g(x) = x \-- 2$ from $\mathbb{R}$ to $\mathbb{R}$, find:
+\begin{enumerate}
+\item $f \circ g$
+ \begin{enumerate}
+ \item[] \answer $f(g(x)) = (x-2)^2 + x - 2 \equiv x^2 - 3x + 2$
+ \end{enumerate}
+
+\item $g \circ f$
+ \begin{enumerate}
+ \item[] \answer $g(f(x)) = x^2 + x - 2$
+ \end{enumerate}
+
+\item $f \circ f$
+ \begin{enumerate}
+ \item[] \answer $f(f(x)) = (x^2 + x)^2 + x \equiv x^4 + 2x^3 + x^2 + x$
+ \end{enumerate}
+
+\item $g \circ g$
+ \begin{enumerate}
+ \item[] \answer $g(g(x)) = x - 4$
+ \end{enumerate}
+
+\end{enumerate}
+
+\textbf{Note:} we have \emph{range}$(f)$ = \emph{range}$(g)$ = \emph{domain}$(f)$ = \emph{domain}$(g)$.
+\
+\end{document}
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diff --git a/hwSat01/HWSAT01DanielShchur.tex~ b/hwSat01/HWSAT01DanielShchur.tex~
deleted file mode 100644
index aa64b3d059b6e7c57d3b24858721cd0d2e0183f6..0000000000000000000000000000000000000000
--- a/hwSat01/HWSAT01DanielShchur.tex~
+++ /dev/null
@@ -1,22 +0,0 @@
-\documentclass[12pt,letterpaper]{article}
-
-\usepackage[table]{xcolor}
-\usepackage{boldline}
-\usepackage{bm}
-\usepackage{forest}
-\usepackage[letterpaper, margin=.75in]{geometry}
-\usepackage{adjustbox}
-
-\newcommand{\problem}{\vspace{.5cm}\textbf{Problem:~~}}
-\newcommand{\answer}{\emph{Answer:~~}}
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-
-\setlength{\parindent}{0pt}
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-
-\title{CSCE 235H SAT Homework 01}
-\author{Daniel Shchur}
diff --git a/hwSat01/SAThw01.pdf b/hwSat01/SAThw01.pdf
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index 0000000000000000000000000000000000000000..3d33ab3698940641274bcd5eee300e5c9239b3c2
--- /dev/null
+++ b/hwSat02/HWSAT02DanielShchur.tex
@@ -0,0 +1,120 @@
+\documentclass[12pt,letterpaper]{article}
+
+\usepackage[table]{xcolor}
+\usepackage{boldline}
+\usepackage{bm}
+\usepackage{forest}
+\usepackage[letterpaper, margin=.75in]{geometry}
+\usepackage{adjustbox}
+\usepackage{graphicx}
+
+\newcommand{\problem}{\vspace{.5cm}\textbf{Problem:~~}}
+\newcommand{\answer}{\emph{Answer:~~}}
+\newcommand{\ansindent}{\indent\hspace{0.5cm}}
+\newcommand{\ansend}{\vspace{0.25cm}\\}
+\newcommand{\rot}[1]{\multicolumn{1}{c}{\adjustbox{angle=90}{#1}}}
+\newcommand{\rotend}[1]{\multicolumn{1}{c V{3}}{\adjustbox{angle=90}{#1}}}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0cm}
+\setlength{\intextsep}{.25cm}
+
+\title{CSCE 235H SAT Homework 02}
+\author{Daniel Shchur}
+
+\begin{document}
+
+\maketitle
+
+\problem Write a CNF formula to model the following scenario:
+
+\begin{enumerate}
+\item[1.] There are four choices of desserts: ice cream, fruit bowl, cake, pie.
+\item[2.] Exactly one dessert must be selected (i.e., one and only one).
+\end{enumerate}
+
+Proceed following the four steps below:
+
+\begin{enumerate}
+\item[1.] First state the propositions and what they represent.
+\item[2.] State the sentence.
+\item[3.] Explain the meaning of the clauses.
+\item[4.] Is the sentence satisfiable? Explain why or why not.
+\end{enumerate}
+
+\answer
+
+\begin{enumerate}
+\item[1.] Proposition D(x) represents ``x is a selected desert'' where:
+ \begin{tabular}{l}
+ 1: Ice cream\\
+ 2: Fruit bowl\\
+ 3: Cake\\
+ 4: Pie\\
+ \end{tabular}
+
+\item[2.]
+ \begin{enumerate}
+ \item[(a)]$\bigvee\limits_{x=1}^{4} P(x)$
+ \item[(b)]$\bigwedge\limits_{x=1}^{3} \bigwedge\limits_{y=x+1}^{4} (\neg P(x) \vee \neg P(y))$
+ \end{enumerate}
+
+\item[3.]
+ \begin{enumerate}
+ \item[(a)]Requires that a desert is selected.
+ \item[(b)]Requires that at most, one desert is selected.
+ \end{enumerate}
+
+\item[4.] This sentence is satisfiable in 4 different ways, where all but one of the variables is false. This makes sense since only one desert must be selected.\\
+ Ex: Ice cream = F, Fruit bowl = F, Cake = F, Pie = T
+
+\end{enumerate}
+
+\newpage
+
+\problem Problem B: Write a CNF formula to model the following scenario:
+
+\begin{enumerate}
+\item[1.] The four states (NE, IA, KS, MO) on the map shown in Figure 1 must be colored using three colors: red, green, and blue.
+
+ \begin{figure}[htb]
+ \centering
+ \includegraphics[scale=0.15]{states}
+ \caption{Four states (NE, IA, KS, MO)}
+ \end{figure}
+
+\item[2.] Each state must be colored with exactly one color.
+\item[3.] Adjacent states (i.e., states sharing a border line) cannot have the same color.
+\end{enumerate}
+
+Proceed following the four steps of Problem A.\ansend
+
+\answer
+
+\begin{enumerate}
+\item[1.] Proposition $S(x,c)$ represents ``A state $x$ has a color $c$'' where:
+ \begin{tabular}{c | c}
+ NE = 1 & R = 1\\
+ IA = 2 & G = 2\\
+ KS = 3 & B = 3\\
+ MO = 4 & \\
+ \end{tabular}
+
+\item[2.]
+ \begin{enumerate}
+ \item[(a)] $\bigwedge\limits_{x=1}^{4} \bigvee\limits_{c=1}^{3} S(x,c)$
+ \item[(b)] $\bigwedge\limits_{x=1}^{2} \bigwedge\limits_{c=1}^{2} \bigwedge\limits_{d=c+1}^{3} (\neg S(x,c) \vee \neg S(x, d))$
+ \item[(c)] $\bigvee\limits_{c=1}^{3} (S(2,c) \wedge S(3, c))$
+ \end{enumerate}
+
+\item[3.]
+ \begin{enumerate}
+ \item[(a)] Requires that every state have a color.
+ \item[(b)] Requires that every state have at most one color.
+ \item[(c)] Requires that states 2 and 3 have the same color as they do not share a border.
+ \end{enumerate}
+\item[4.] Yes, the problem is satisfiable as long as KS and IA are the same colors. All borders overlap except for those borders. This presents with 3 different solutions solved in the same manner. Where IA and KS are the same color and the other two states are the other two colors.\\ Ex: NE=R, IA=B, KS=B, MO=G
+\end{enumerate}
+
+
+\end{document}
\ No newline at end of file
diff --git a/hwSat02/SAThw02.pdf b/hwSat02/SAThw02.pdf
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diff --git a/hwSat03/HWSAT03DanielShchur.pdf b/hwSat03/HWSAT03DanielShchur.pdf
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diff --git a/hwSat03/HWSAT03DanielShchur.tex b/hwSat03/HWSAT03DanielShchur.tex
new file mode 100644
index 0000000000000000000000000000000000000000..60f5a03124a750673fb7ab2cd8db2d065816bcc5
--- /dev/null
+++ b/hwSat03/HWSAT03DanielShchur.tex
@@ -0,0 +1,128 @@
+\documentclass[12pt,letterpaper]{article}
+
+\usepackage[letterpaper, margin=.75in]{geometry}
+
+\newcommand{\problem[1]}{\vspace{.5cm}\textbf{Problem {#1}:~~}}
+\newcommand{\answer}{\emph{Answer:~~}}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0cm}
+\setlength{\intextsep}{.25cm}
+
+\title{CSCE 235H SAT Homework 03}
+\author{Daniel Shchur}
+
+\begin{document}
+
+\maketitle
+
+\problem[A]
+
+\begin{itemize}
+\item Select a \emph{satisfiable} benchmark instance from the SATLIB website.
+\item Run the instance through MiniSAT.
+\item Submit the generated output as well as the contents of the results file.
+\end{itemize}
+
+\footnotesize
+\begin{verbatim}
+*** Console ***
+** File: uf125-0100.cnf **
+
+WARNING: for repeatability, setting FPU to use double precision
+============================[ Problem Statistics ]=============================
+| |
+| Number of variables: 125 |
+| Number of clauses: 538 |
+| Parse time: 0.00 s |
+| |
+============================[ Search Statistics ]==============================
+| Conflicts | ORIGINAL | LEARNT | Progress |
+| | Vars Clauses Literals | Limit Clauses Lit/Cl | |
+===============================================================================
+| 100 | 125 538 1614 | 197 100 10 | 0.006 % |
+| 250 | 125 538 1614 | 216 149 8 | 0.007 % |
+===============================================================================
+restarts : 4
+conflicts : 470 (49025 /sec)
+decisions : 617 (0.00 % random) (64358 /sec)
+propagations : 13417 (1399499 /sec)
+conflict literals : 3957 (18.68 % deleted)
+Memory used : 20.00 MB
+CPU time : 0.009587 s
+
+SATISFIABLE
+
+*** Ouput.txt ***
+
+SAT
+-1 2 3 -4 -5 -6 -7 -8 9 -10 -11 -12 -13 14 -15 16 17 -18 19 20 -21 -22 23 -24 25
+-26 27 28 -29 -30 31 -32 33 -34 -35 -36 37 -38 39 40 41 -42 -43 -44 -45 46 47 48
+49 -50 -51 -52 -53 -54 -55 56 57 58 59 60 61 -62 63 -64 65 66 67 68 -69 70 71 72
+-73 -74 75 -76 77 -78 -79 80 -81 82 -83 84 85 -86 87 -88 -89 90 -91 -92 -93 94
+-95 -96 -97 -98 -99 -100 -101 102 103 -104 -105 106 -107 -108 -109 110 -111 -112
+-113 -114 115 116 -117 -118 119 120 -121 122 123 -124 -125 0
+\end{verbatim}
+\normalsize
+\newpage
+
+\problem[B]
+
+\begin{itemize}
+\item Select an \emph{unsatisfiable} benchmark instance from the SATLIB website.
+\item Run the instance through MiniSAT.
+\item Submit the generated output as well as the contents of the results file.
+\end{itemize}
+
+\footnotesize
+\begin{verbatim}
+*** Console ***
+** File: uuf150-0100.cnf **
+
+WARNING: for repeatability, setting FPU to use double precision
+============================[ Problem Statistics ]=============================
+| |
+| Number of variables: 150 |
+| Number of clauses: 645 |
+| Parse time: 0.00 s |
+| |
+============================[ Search Statistics ]==============================
+| Conflicts | ORIGINAL | LEARNT | Progress |
+| | Vars Clauses Literals | Limit Clauses Lit/Cl | |
+===============================================================================
+| 100 | 150 645 1935 | 236 100 10 | 0.004 % |
+| 250 | 150 645 1935 | 260 250 9 | 0.004 % |
+| 475 | 150 645 1935 | 286 205 9 | 0.004 % |
+| 812 | 150 645 1935 | 314 246 8 | 0.004 % |
+| 1318 | 150 645 1935 | 346 271 8 | 0.004 % |
+| 2077 | 150 645 1935 | 380 319 7 | 0.004 % |
+| 3216 | 150 645 1935 | 418 287 6 | 0.004 % |
+| 4924 | 150 645 1935 | 460 273 6 | 0.004 % |
+===============================================================================
+restarts : 30
+conflicts : 6099 (108221 /sec)
+decisions : 7266 (0.00 % random) (128928 /sec)
+propagations : 193502 (3433504 /sec)
+conflict literals : 50578 (24.72 % deleted)
+Memory used : 20.00 MB
+CPU time : 0.056357 s
+
+UNSATISFIABLE
+
+*** Output.txt ***
+
+UNSAT
+\end{verbatim}
+\vspace{-0.3cm}
+\normalsize
+
+\problem[C]
+
+\begin{itemize}
+\item Select a benchmark from the SATLIB website.
+\item Read its description page.
+\item Briefly describe, in your own words, what the benchmark is modeling.
+\end{itemize}
+
+\answer SAT-encoded "Morphed" Graph Coloring Problem: The benchmak models a problem from graph theory where a graph/map is made of verticies and edges and all connected vertices always have different colors. The goal for the SAT model is to decide whether for a particular number of colors, a coloring of the given graph exists.
+\end{document}
\ No newline at end of file
diff --git a/hwSat03/SAThw03.pdf b/hwSat03/SAThw03.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..068120d19661224e12404d2819813636a27932ef
Binary files /dev/null and b/hwSat03/SAThw03.pdf differ
diff --git a/hwSat03/problemA/consoleProbA.txt b/hwSat03/problemA/consoleProbA.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2f0716765838baeba9eb7f3dd4bba480c4928b09
--- /dev/null
+++ b/hwSat03/problemA/consoleProbA.txt
@@ -0,0 +1,23 @@
+WARNING: for repeatability, setting FPU to use double precision
+============================[ Problem Statistics ]=============================
+| |
+| Number of variables: 125 |
+| Number of clauses: 538 |
+| Parse time: 0.00 s |
+| |
+============================[ Search Statistics ]==============================
+| Conflicts | ORIGINAL | LEARNT | Progress |
+| | Vars Clauses Literals | Limit Clauses Lit/Cl | |
+===============================================================================
+| 100 | 125 538 1614 | 197 100 10 | 0.006 % |
+| 250 | 125 538 1614 | 216 149 8 | 0.007 % |
+===============================================================================
+restarts : 4
+conflicts : 470 (49025 /sec)
+decisions : 617 (0.00 % random) (64358 /sec)
+propagations : 13417 (1399499 /sec)
+conflict literals : 3957 (18.68 % deleted)
+Memory used : 20.00 MB
+CPU time : 0.009587 s
+
+SATISFIABLE
diff --git a/hwSat03/problemA/outputProbA.txt b/hwSat03/problemA/outputProbA.txt
new file mode 100644
index 0000000000000000000000000000000000000000..914e80825009bec4d3e6f3e488d04d5bcc19165f
--- /dev/null
+++ b/hwSat03/problemA/outputProbA.txt
@@ -0,0 +1,2 @@
+SAT
+-1 2 3 -4 -5 -6 -7 -8 9 -10 -11 -12 -13 14 -15 16 17 -18 19 20 -21 -22 23 -24 25 -26 27 28 -29 -30 31 -32 33 -34 -35 -36 37 -38 39 40 41 -42 -43 -44 -45 46 47 48 49 -50 -51 -52 -53 -54 -55 56 57 58 59 60 61 -62 63 -64 65 66 67 68 -69 70 71 72 -73 -74 75 -76 77 -78 -79 80 -81 82 -83 84 85 -86 87 -88 -89 90 -91 -92 -93 94 -95 -96 -97 -98 -99 -100 -101 102 103 -104 -105 106 -107 -108 -109 110 -111 -112 -113 -114 115 116 -117 -118 119 120 -121 122 123 -124 -125 0
diff --git a/hwSat03/problemA/uf125-0100.cnf b/hwSat03/problemA/uf125-0100.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..7d027bbe858c76f0c37b6e2eef60fa7b7cd1e3be
--- /dev/null
+++ b/hwSat03/problemA/uf125-0100.cnf
@@ -0,0 +1,546 @@
+c This Formular is generated by mcnf
+c
+c horn? no
+c forced? no
+c mixed sat? no
+c clause length = 3
+c
+p cnf 125 538
+ 61 4 2 0
+-86 116 90 0
+98 75 62 0
+-49 94 62 0
+-99 -90 -59 0
+18 30 -107 0
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diff --git a/hwSat03/problemB/consoleProbB.txt b/hwSat03/problemB/consoleProbB.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4d8a1ee564705a5bf39c419c58e96d4122c3df60
--- /dev/null
+++ b/hwSat03/problemB/consoleProbB.txt
@@ -0,0 +1,29 @@
+WARNING: for repeatability, setting FPU to use double precision
+============================[ Problem Statistics ]=============================
+| |
+| Number of variables: 150 |
+| Number of clauses: 645 |
+| Parse time: 0.00 s |
+| |
+============================[ Search Statistics ]==============================
+| Conflicts | ORIGINAL | LEARNT | Progress |
+| | Vars Clauses Literals | Limit Clauses Lit/Cl | |
+===============================================================================
+| 100 | 150 645 1935 | 236 100 10 | 0.004 % |
+| 250 | 150 645 1935 | 260 250 9 | 0.004 % |
+| 475 | 150 645 1935 | 286 205 9 | 0.004 % |
+| 812 | 150 645 1935 | 314 246 8 | 0.004 % |
+| 1318 | 150 645 1935 | 346 271 8 | 0.004 % |
+| 2077 | 150 645 1935 | 380 319 7 | 0.004 % |
+| 3216 | 150 645 1935 | 418 287 6 | 0.004 % |
+| 4924 | 150 645 1935 | 460 273 6 | 0.004 % |
+===============================================================================
+restarts : 30
+conflicts : 6099 (108221 /sec)
+decisions : 7266 (0.00 % random) (128928 /sec)
+propagations : 193502 (3433504 /sec)
+conflict literals : 50578 (24.72 % deleted)
+Memory used : 20.00 MB
+CPU time : 0.056357 s
+
+UNSATISFIABLE
diff --git a/hwSat03/problemB/outputProbB.txt b/hwSat03/problemB/outputProbB.txt
new file mode 100644
index 0000000000000000000000000000000000000000..abb3248efb654890006fa920d31b28cdb5585d38
--- /dev/null
+++ b/hwSat03/problemB/outputProbB.txt
@@ -0,0 +1 @@
+UNSAT
diff --git a/hwSat03/problemB/uuf150-0100.cnf b/hwSat03/problemB/uuf150-0100.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..14df42ac86aa604459c7b5ff734d54f360d7fcb5
--- /dev/null
+++ b/hwSat03/problemB/uuf150-0100.cnf
@@ -0,0 +1,653 @@
+c This Formular is generated by mcnf
+c
+c horn? no
+c forced? no
+c mixed sat? no
+c clause length = 3
+c
+p cnf 150 645
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+-21 9 129 0
+-137 -128 -149 0
+51 72 126 0
+-12 -61 45 0
+8 134 17 0
+-18 -48 55 0
+62 -116 46 0
+14 98 140 0
+79 67 -104 0
+19 38 5 0
+76 -112 106 0
+133 57 -69 0
+57 -35 42 0
+105 30 114 0
+46 -59 -57 0
+101 -42 83 0
+-80 -97 -149 0
+-45 134 91 0
+115 145 -102 0
+64 70 -47 0
+73 -123 -53 0
+-132 76 131 0
+-15 -10 -57 0
+-150 59 -83 0
+-98 128 16 0
diff --git a/hwSat03/uuf150-645.tar.gz b/hwSat03/uuf150-645.tar.gz
new file mode 100644
index 0000000000000000000000000000000000000000..141f5a85d3e5b5e3c1d648a903e59fdbd5f91323
Binary files /dev/null and b/hwSat03/uuf150-645.tar.gz differ
diff --git a/hwSat04/.classpath b/hwSat04/.classpath
new file mode 100644
index 0000000000000000000000000000000000000000..fb5011632c0ab8d6649a148c6fb5845a1b34c747
--- /dev/null
+++ b/hwSat04/.classpath
@@ -0,0 +1,6 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<classpath>
+ <classpathentry kind="src" path="src"/>
+ <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER"/>
+ <classpathentry kind="output" path="bin"/>
+</classpath>
diff --git a/hwSat04/.gitignore b/hwSat04/.gitignore
new file mode 100644
index 0000000000000000000000000000000000000000..ae3c1726048cd06b9a143e0376ed46dd9b9a8d53
--- /dev/null
+++ b/hwSat04/.gitignore
@@ -0,0 +1 @@
+/bin/
diff --git a/hwSat04/.project b/hwSat04/.project
new file mode 100644
index 0000000000000000000000000000000000000000..cdd791261408266d9b8615eab77b66dee863956d
--- /dev/null
+++ b/hwSat04/.project
@@ -0,0 +1,17 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<projectDescription>
+ <name>hwSat04</name>
+ <comment></comment>
+ <projects>
+ </projects>
+ <buildSpec>
+ <buildCommand>
+ <name>org.eclipse.jdt.core.javabuilder</name>
+ <arguments>
+ </arguments>
+ </buildCommand>
+ </buildSpec>
+ <natures>
+ <nature>org.eclipse.jdt.core.javanature</nature>
+ </natures>
+</projectDescription>
diff --git a/hwSat04/SAThw04.pdf b/hwSat04/SAThw04.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..370402201119f620cc4655ceb8489f3a0bd21ac1
Binary files /dev/null and b/hwSat04/SAThw04.pdf differ
diff --git a/hwSat04/problemA/dessert.cnf b/hwSat04/problemA/dessert.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..f674d769f832c64a02b76b01250392e10b5f6179
--- /dev/null
+++ b/hwSat04/problemA/dessert.cnf
@@ -0,0 +1,20 @@
+c Daniel Shchur
+c
+c Dessert CNF Problem
+c
+c Only one desert must be chosen
+c
+c Variables:
+c 1: icecream
+c 2: fruitbowl
+c 3: cake
+c 4: pie
+c
+p cnf 4 7
+1 2 3 4 0
+-1 -2 0
+-1 -3 0
+-1 -4 0
+-2 -3 0
+-2 -4 0
+-3 -4 0
\ No newline at end of file
diff --git a/hwSat04/problemA/results.txt b/hwSat04/problemA/results.txt
new file mode 100644
index 0000000000000000000000000000000000000000..66326886c5ea8cb88411f4ffebbb1b98c0b0d64d
--- /dev/null
+++ b/hwSat04/problemA/results.txt
@@ -0,0 +1,2 @@
+SAT
+-1 2 -3 -4 0
diff --git a/hwSat04/src/GenerateSudoku.java b/hwSat04/src/GenerateSudoku.java
new file mode 100644
index 0000000000000000000000000000000000000000..1fce3e65145511214759d0da43ab027a8e101dc2
--- /dev/null
+++ b/hwSat04/src/GenerateSudoku.java
@@ -0,0 +1,29 @@
+/*
+ * Daniel Shchur
+ * CSCE235H
+ *
+ * Sudoku CNF DIMACS Generator
+ *
+ * Description: Generates the Sudoku CNF problem in DIMACS format to be able to be used in a minisat solver
+ *
+ * NOTE: Current Version only sets up the CNF file, it does NOT run on minisat.
+ */
+
+public class GenerateSudoku {
+
+ public static void main(String[] args) {
+
+ //Iterates over 3 loops so that it goes through every grid in the sudoku board giving a number 1-9
+ for(int i=1; i < 10; i++) {
+ for(int n=1; n < 10; n++) {
+ for(int j=1; j < 10; j++) {
+ //Prints the combination as inj
+ System.out.print(" " + i + n + j);
+ }
+ //Prints the terminating character
+ System.out.println(" 0");
+ }
+
+ }
+ }
+}
\ No newline at end of file
diff --git a/hwSat05/.classpath b/hwSat05/.classpath
new file mode 100644
index 0000000000000000000000000000000000000000..0cbf9cda0437f562541924448019c8381753a991
--- /dev/null
+++ b/hwSat05/.classpath
@@ -0,0 +1,6 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<classpath>
+ <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER"/>
+ <classpathentry kind="src" path="src"/>
+ <classpathentry kind="output" path="bin"/>
+</classpath>
diff --git a/hwSat05/.gitignore b/hwSat05/.gitignore
new file mode 100644
index 0000000000000000000000000000000000000000..38cd1300081480f1e8082498bd553874b364ecad
--- /dev/null
+++ b/hwSat05/.gitignore
@@ -0,0 +1,2 @@
+/GenerateSudoku.class
+/bin/
diff --git a/hwSat05/.project b/hwSat05/.project
new file mode 100644
index 0000000000000000000000000000000000000000..85e4d749ddf243bb5f930b8adf21c098f74d214b
--- /dev/null
+++ b/hwSat05/.project
@@ -0,0 +1,17 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<projectDescription>
+ <name>hwSat05</name>
+ <comment></comment>
+ <projects>
+ </projects>
+ <buildSpec>
+ <buildCommand>
+ <name>org.eclipse.jdt.core.javabuilder</name>
+ <arguments>
+ </arguments>
+ </buildCommand>
+ </buildSpec>
+ <natures>
+ <nature>org.eclipse.jdt.core.javanature</nature>
+ </natures>
+</projectDescription>
diff --git a/hwSat05/SAThw05.pdf b/hwSat05/SAThw05.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..a2166b8652f2bc418770647b8a06db9288be6d53
Binary files /dev/null and b/hwSat05/SAThw05.pdf differ
diff --git a/hwSat05/emptySudoku.cnf b/hwSat05/emptySudoku.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..2dd61d3e72f2428019b5765f651e23e1842a3906
--- /dev/null
+++ b/hwSat05/emptySudoku.cnf
@@ -0,0 +1,3170 @@
+c Daniel Shchur
+c
+c Sudoku CNF Problem
+c
+c Provides an empty Sudoku board to be solved
+c
+p cnf 729 3159
+c (1) Every row contains every number:
+111 121 131 141 151 161 171 181 191 0
+112 122 132 142 152 162 172 182 192 0
+113 123 133 143 153 163 173 183 193 0
+114 124 134 144 154 164 174 184 194 0
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+119 129 139 149 159 169 179 189 199 0
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+918 928 938 948 958 968 978 988 998 0
+919 929 939 949 959 969 979 989 999 0
+c (2) Every column contains every number:
+111 211 311 411 511 611 711 811 911 0
+112 212 312 412 512 612 712 812 912 0
+113 213 313 413 513 613 713 813 913 0
+114 214 314 414 514 614 714 814 914 0
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+155 255 355 455 555 655 755 855 955 0
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+197 297 397 497 597 697 797 897 997 0
+198 298 398 498 598 698 798 898 998 0
+199 299 399 499 599 699 799 899 999 0
+c (3) Each 3x3 block contains every number:
+111 121 131 211 221 231 311 321 331 0
+112 122 132 212 222 232 312 322 332 0
+113 123 133 213 223 233 313 323 333 0
+114 124 134 214 224 234 314 324 334 0
+115 125 135 215 225 235 315 325 335 0
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+413 423 433 513 523 533 613 623 633 0
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+711 721 731 811 821 831 911 921 931 0
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+719 729 739 819 829 839 919 929 939 0
+741 751 761 841 851 861 941 951 961 0
+742 752 762 842 852 862 942 952 962 0
+743 753 763 843 853 863 943 953 963 0
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+778 788 798 878 888 898 978 988 998 0
+779 789 799 879 889 899 979 989 999 0
+c (4) No cell contains more than one number:
+-111 -112 0
+-111 -113 0
+-111 -114 0
+-111 -115 0
+-111 -116 0
+-111 -117 0
+-111 -118 0
+-111 -119 0
+-112 -113 0
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\ No newline at end of file
diff --git a/hwSat05/emptySudokuResults.txt b/hwSat05/emptySudokuResults.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0d78b0ec72441fdd73ba7dddc1d123f1220f6956
--- /dev/null
+++ b/hwSat05/emptySudokuResults.txt
@@ -0,0 +1,2 @@
+SAT
+-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24 -25 -26 -27 -28 -29 -30 -31 -32 -33 -34 -35 -36 -37 -38 -39 -40 -41 -42 -43 -44 -45 -46 -47 -48 -49 -50 -51 -52 -53 -54 -55 -56 -57 -58 -59 -60 -61 -62 -63 -64 -65 -66 -67 -68 -69 -70 -71 -72 -73 -74 -75 -76 -77 -78 -79 -80 -81 -82 -83 -84 -85 -86 -87 -88 -89 -90 -91 -92 -93 -94 -95 -96 -97 -98 -99 -100 -101 -102 -103 -104 -105 -106 -107 -108 -109 -110 -111 -112 -113 -114 -115 -116 -117 -118 119 -120 -121 -122 -123 -124 -125 -126 -127 128 -129 -130 -131 -132 -133 134 -135 -136 -137 -138 -139 -140 -141 -142 143 -144 -145 -146 -147 -148 -149 -150 -151 -152 -153 -154 -155 -156 157 -158 -159 -160 161 -162 -163 -164 -165 -166 -167 -168 -169 -170 -171 -172 -173 -174 -175 176 -177 -178 -179 -180 -181 182 -183 -184 -185 -186 -187 -188 -189 -190 -191 -192 -193 -194 195 -196 -197 -198 -199 -200 -201 -202 -203 -204 -205 -206 -207 -208 -209 -210 -211 212 -213 -214 -215 -216 -217 -218 -219 -220 -221 -222 -223 -224 -225 -226 227 -228 -229 -230 -231 -232 -233 -234 -235 236 -237 -238 -239 -240 -241 -242 -243 -244 245 -246 -247 -248 -249 -250 -251 -252 -253 -254 -255 -256 -257 -258 259 -260 -261 -262 -263 -264 -265 -266 -267 268 -269 -270 271 -272 -273 -274 -275 -276 -277 -278 -279 -280 -281 -282 -283 284 -285 -286 -287 -288 -289 -290 -291 -292 293 -294 -295 -296 -297 -298 -299 -300 -301 -302 -303 -304 -305 -306 -307 -308 -309 -310 -311 -312 313 -314 -315 -316 -317 -318 -319 -320 -321 -322 -323 -324 325 -326 -327 -328 -329 -330 331 -332 -333 -334 -335 -336 -337 -338 -339 -340 -341 -342 -343 -344 -345 346 -347 -348 -349 -350 -351 352 -353 -354 -355 -356 -357 -358 -359 -360 -361 -362 -363 364 -365 -366 -367 -368 -369 -370 -371 -372 -373 -374 -375 -376 -377 378 -379 -380 -381 -382 -383 -384 -385 -386 -387 -388 389 -390 -391 -392 -393 -394 -395 -396 397 -398 -399 -400 -401 -402 -403 -404 -405 -406 -407 -408 -409 -410 411 -412 -413 -414 -415 -416 -417 -418 -419 -420 -421 -422 -423 424 -425 -426 -427 -428 -429 -430 -431 -432 -433 -434 -435 -436 -437 -438 439 -440 -441 -442 -443 -444 -445 -446 447 -448 -449 -450 -451 -452 -453 -454 -455 -456 -457 458 -459 -460 -461 -462 463 -464 -465 -466 -467 -468 -469 -470 -471 472 -473 -474 -475 -476 -477 -478 -479 -480 -481 -482 -483 -484 485 -486 -487 -488 -489 -490 -491 -492 -493 -494 -495 496 -497 -498 -499 -500 -501 -502 -503 -504 -505 -506 -507 -508 -509 -510 -511 -512 -513 -514 -515 -516 -517 518 -519 -520 -521 -522 523 -524 -525 -526 -527 -528 -529 -530 -531 -532 -533 -534 535 -536 -537 -538 -539 -540 -541 542 -543 -544 -545 -546 -547 -548 -549 -550 -551 -552 -553 554 -555 -556 -557 -558 -559 -560 -561 -562 -563 -564 -565 566 -567 -568 -569 -570 -571 -572 -573 -574 -575 -576 577 -578 -579 -580 581 -582 -583 -584 -585 -586 -587 -588 -589 -590 -591 -592 -593 -594 -595 -596 -597 -598 599 -600 -601 -602 -603 -604 -605 -606 -607 -608 -609 -610 -611 -612 -613 -614 -615 616 -617 -618 -619 -620 -621 622 -623 -624 -625 -626 -627 -628 -629 -630 -631 -632 -633 -634 -635 -636 637 -638 -639 -640 641 -642 -643 -644 -645 -646 -647 -648 -649 -650 -651 -652 -653 -654 655 -656 -657 -658 -659 -660 -661 -662 -663 -664 -665 -666 -667 -668 669 -670 -671 -672 673 -674 -675 -676 -677 -678 -679 -680 -681 -682 -683 -684 -685 -686 -687 688 -689 -690 -691 -692 -693 694 -695 -696 -697 -698 -699 -700 -701 -702 -703 -704 -705 -706 -707 -708 -709 -710 -711 -712 -713 714 -715 -716 -717 -718 -719 -720 -721 -722 -723 -724 -725 -726 -727 -728 729 -730 -731 732 -733 -734 -735 -736 -737 -738 -739 -740 -741 -742 -743 -744 -745 -746 -747 748 -749 -750 -751 -752 -753 -754 -755 756 -757 -758 -759 -760 -761 -762 -763 -764 -765 -766 767 -768 -769 -770 -771 -772 -773 -774 775 -776 -777 -778 -779 -780 -781 -782 783 -784 -785 -786 -787 -788 -789 -790 791 -792 -793 -794 -795 -796 -797 -798 -799 -800 -801 -802 -803 -804 -805 -806 -807 -808 -809 -810 -811 -812 -813 -814 -815 -816 817 -818 -819 -820 821 -822 -823 -824 -825 -826 -827 -828 -829 -830 -831 -832 -833 -834 -835 -836 -837 838 -839 -840 -841 -842 -843 844 -845 -846 -847 -848 -849 -850 -851 -852 853 -854 -855 -856 -857 -858 -859 -860 -861 -862 -863 -864 865 -866 -867 -868 -869 -870 -871 -872 -873 -874 -875 -876 -877 -878 879 -880 -881 -882 -883 -884 -885 886 -887 -888 -889 -890 -891 892 -893 -894 -895 -896 -897 -898 -899 -900 -901 -902 -903 -904 -905 -906 -907 -908 -909 -910 -911 -912 -913 -914 915 -916 -917 -918 -919 -920 -921 -922 -923 -924 -925 926 -927 -928 -929 -930 -931 -932 933 -934 -935 -936 -937 -938 -939 -940 -941 -942 -943 -944 -945 -946 -947 -948 949 -950 951 -952 -953 -954 -955 -956 -957 -958 -959 -960 -961 962 -963 -964 -965 -966 -967 -968 -969 -970 -971 -972 -973 974 -975 -976 -977 -978 -979 -980 -981 -982 -983 -984 -985 -986 987 -988 -989 -990 -991 -992 -993 -994 -995 -996 -997 998 -999 0
diff --git a/hwSat05/src/GenerateSudoku.java b/hwSat05/src/GenerateSudoku.java
new file mode 100644
index 0000000000000000000000000000000000000000..e3a090c93b76e9c8ccb75cd1f20aab9d1b13fa40
--- /dev/null
+++ b/hwSat05/src/GenerateSudoku.java
@@ -0,0 +1,86 @@
+/*
+ * Daniel Shchur
+ * CSCE235H
+ *
+ * Sudoku CNF DIMACS Generator
+ *
+ * Description: Generates the Sudoku CNF problem in DIMACS format to be able to be used in a minisat solver
+ *
+ * NOTE: Current Version generates the problem for an empty sudoku board
+ */
+
+public class GenerateSudoku {
+
+ public static void main(String[] args) {
+
+ //Print statement to generate the comment section.
+ System.out.print(
+ "c Daniel Shchur\n" +
+ "c\n" +
+ "c Sudoku CNF Problem\n" +
+ "c\n" +
+ "c Provides an empty Sudoku board to be solved\n" +
+ "c\n");
+
+ //Prints header
+ System.out.println("p cnf 729 3159");
+
+ //Iterates over 3 loops so that it goes through every row in the sudoku board giving a number 1-9
+ System.out.println("c (1) Every row contains every number:");
+ for(int i=1; i < 10; i++) {
+ for(int n=1; n < 10; n++) {
+ for(int j=1; j < 10; j++) {
+ //Prints the combination as a single variable
+ System.out.printf("%d%d%d ", i, j, n);
+ }
+ //Prints the terminating character
+ System.out.println("0");
+ }
+ }
+
+ //Iterates over 3 loops so that it goes through every column in the sudoku board giving a number 1-9
+ System.out.println("c (2) Every column contains every number:");
+ for(int j=1; j < 10; j++) {
+ for(int n=1; n < 10; n++) {
+ for(int i=1; i < 10; i++) {
+ //Prints the combination as a single variable
+ System.out.printf("%d%d%d ", i, j, n);
+ }
+ //Prints the terminating character
+ System.out.println("0");
+ }
+ }
+
+ //Iterates over 5 loops so that it goes through every 3x3 grid in the sudoku board giving a number 1-9
+ System.out.println("c (3) Each 3x3 block contains every number:");
+ for(int r=0; r < 3; r++) {
+ for(int s=0; s < 3; s++) {
+ for(int n=1; n < 10; n++) {
+ for(int i=1; i < 4; i++) {
+ for(int j=1; j < 4; j++) {
+ //Prints the combination as a single variable
+ System.out.printf("%d%d%d ", (3*r + i), (3*s + j), n);
+ }
+ }
+ //Prints the terminating character
+ System.out.println("0");
+ }
+ }
+ }
+
+ //Iterates over 4 loops so that it goes through every cell in the sudoku board making sure no cell has more than 1 number
+ System.out.println("c (4) No cell contains more than one number:");
+ for(int i=1; i < 10; i++) {
+ for(int j=1; j < 10; j++) {
+ for(int n=1; n < 9; n++) {
+ for(int m=n+1; m < 10; m++) {
+ //Prints the combination as a single variable with terminating character
+ System.out.printf("%d%d%d %d%d%d %d\n", -i, j, n, -i, j, m, 0);
+ }
+ }
+ }
+ }
+
+ //End of code
+ }
+}
\ No newline at end of file
diff --git a/hwSat06/.classpath b/hwSat06/.classpath
new file mode 100644
index 0000000000000000000000000000000000000000..0cbf9cda0437f562541924448019c8381753a991
--- /dev/null
+++ b/hwSat06/.classpath
@@ -0,0 +1,6 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<classpath>
+ <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER"/>
+ <classpathentry kind="src" path="src"/>
+ <classpathentry kind="output" path="bin"/>
+</classpath>
diff --git a/hwSat06/.gitignore b/hwSat06/.gitignore
new file mode 100644
index 0000000000000000000000000000000000000000..38cd1300081480f1e8082498bd553874b364ecad
--- /dev/null
+++ b/hwSat06/.gitignore
@@ -0,0 +1,2 @@
+/GenerateSudoku.class
+/bin/
diff --git a/hwSat06/.project b/hwSat06/.project
new file mode 100644
index 0000000000000000000000000000000000000000..750740fd58803c59ee4b6c0bfcb1d1872add5f60
--- /dev/null
+++ b/hwSat06/.project
@@ -0,0 +1,17 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<projectDescription>
+ <name>hwSat06</name>
+ <comment></comment>
+ <projects>
+ </projects>
+ <buildSpec>
+ <buildCommand>
+ <name>org.eclipse.jdt.core.javabuilder</name>
+ <arguments>
+ </arguments>
+ </buildCommand>
+ </buildSpec>
+ <natures>
+ <nature>org.eclipse.jdt.core.javanature</nature>
+ </natures>
+</projectDescription>
diff --git a/hwSat06/SAThw06.pdf b/hwSat06/SAThw06.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e6fde77a4c64956b0d6a51f15c94452bbcea367d
Binary files /dev/null and b/hwSat06/SAThw06.pdf differ
diff --git a/hwSat06/aiEscargot.cnf b/hwSat06/aiEscargot.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..63ebc442e4fd9f139f820bda4a04bc5f6dd757ec
--- /dev/null
+++ b/hwSat06/aiEscargot.cnf
@@ -0,0 +1,3195 @@
+c Daniel Shchur
+c
+c Sudoku CNF Problem
+c
+c Provides a Sudoku board to be solved
+c
+p cnf 729 3183
+c (1) Every row contains every number:
+111 121 131 141 151 161 171 181 191 0
+112 122 132 142 152 162 172 182 192 0
+113 123 133 143 153 163 173 183 193 0
+114 124 134 144 154 164 174 184 194 0
+115 125 135 145 155 165 175 185 195 0
+116 126 136 146 156 166 176 186 196 0
+117 127 137 147 157 167 177 187 197 0
+118 128 138 148 158 168 178 188 198 0
+119 129 139 149 159 169 179 189 199 0
+211 221 231 241 251 261 271 281 291 0
+212 222 232 242 252 262 272 282 292 0
+213 223 233 243 253 263 273 283 293 0
+214 224 234 244 254 264 274 284 294 0
+215 225 235 245 255 265 275 285 295 0
+216 226 236 246 256 266 276 286 296 0
+217 227 237 247 257 267 277 287 297 0
+218 228 238 248 258 268 278 288 298 0
+219 229 239 249 259 269 279 289 299 0
+311 321 331 341 351 361 371 381 391 0
+312 322 332 342 352 362 372 382 392 0
+313 323 333 343 353 363 373 383 393 0
+314 324 334 344 354 364 374 384 394 0
+315 325 335 345 355 365 375 385 395 0
+316 326 336 346 356 366 376 386 396 0
+317 327 337 347 357 367 377 387 397 0
+318 328 338 348 358 368 378 388 398 0
+319 329 339 349 359 369 379 389 399 0
+411 421 431 441 451 461 471 481 491 0
+412 422 432 442 452 462 472 482 492 0
+413 423 433 443 453 463 473 483 493 0
+414 424 434 444 454 464 474 484 494 0
+415 425 435 445 455 465 475 485 495 0
+416 426 436 446 456 466 476 486 496 0
+417 427 437 447 457 467 477 487 497 0
+418 428 438 448 458 468 478 488 498 0
+419 429 439 449 459 469 479 489 499 0
+511 521 531 541 551 561 571 581 591 0
+512 522 532 542 552 562 572 582 592 0
+513 523 533 543 553 563 573 583 593 0
+514 524 534 544 554 564 574 584 594 0
+515 525 535 545 555 565 575 585 595 0
+516 526 536 546 556 566 576 586 596 0
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+c (5) clauses generated from input
+111 0
+167 0
+189 0
+223 0
+252 0
+298 0
+339 0
+346 0
+375 0
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+443 0
+479 0
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diff --git a/hwSat06/aiEscargot.txt b/hwSat06/aiEscargot.txt
new file mode 100644
index 0000000000000000000000000000000000000000..53eec258a336d6952d2dfe3dcfb95061b35f310e
--- /dev/null
+++ b/hwSat06/aiEscargot.txt
@@ -0,0 +1 @@
+1....7.9..3..2...8..96..5....53..9...1..8...26....4...3......1..41.....7..7...3..
\ No newline at end of file
diff --git a/hwSat06/aiEscargotResults.txt b/hwSat06/aiEscargotResults.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3d9b54c4162bcda04a3191bd6330ec6fcbdc606c
--- /dev/null
+++ b/hwSat06/aiEscargotResults.txt
@@ -0,0 +1,2 @@
+SAT
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diff --git a/hwSat06/easy.cnf b/hwSat06/easy.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..b7066512db1e8afa69d150bd159b3d54095a8ac4
--- /dev/null
+++ b/hwSat06/easy.cnf
@@ -0,0 +1,3203 @@
+c Daniel Shchur
+c
+c Sudoku CNF Problem
+c
+c Provides a Sudoku board to be solved
+c
+p cnf 729 3191
+c (1) Every row contains every number:
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+918 928 938 948 958 968 978 988 998 0
+919 929 939 949 959 969 979 989 999 0
+c (2) Every column contains every number:
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+c (3) Each 3x3 block contains every number:
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+779 789 799 879 889 899 979 989 999 0
+c (4) No cell contains more than one number:
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diff --git a/hwSat06/easy.txt b/hwSat06/easy.txt
new file mode 100644
index 0000000000000000000000000000000000000000..36fd63fe43a222fb57fa8a308080d4565f3d8817
--- /dev/null
+++ b/hwSat06/easy.txt
@@ -0,0 +1 @@
+..3.2.6..9..3.5..1..18.64....81.29..7.......8..67.82....26.95..8..2.3..9..5.1.3..
\ No newline at end of file
diff --git a/hwSat06/easyResults.txt b/hwSat06/easyResults.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f9fd141753c2623a9eba65cdbb3801ef7006c1d1
--- /dev/null
+++ b/hwSat06/easyResults.txt
@@ -0,0 +1,2 @@
+SAT
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diff --git a/hwSat06/src/GenerateSudoku.java b/hwSat06/src/GenerateSudoku.java
new file mode 100644
index 0000000000000000000000000000000000000000..2a52f9c075060aa7bf8029697c906a3cfa8f87c3
--- /dev/null
+++ b/hwSat06/src/GenerateSudoku.java
@@ -0,0 +1,168 @@
+import java.util.Scanner;
+import java.util.regex.Pattern;
+
+/*
+ * Daniel Shchur
+ * CSCE235H
+ *
+ * Sudoku CNF DIMACS Generator
+ *
+ * Description: Generates the Sudoku CNF problem in DIMACS format to be able to be used in a minisat solver
+ *
+ * Current Version generates the problem according to input string through STDIN.
+ */
+
+public class GenerateSudoku {
+
+ public static void main(String[] args) {
+
+ //scanner to read input from stdinput.
+ String input = null;
+ Scanner in = new Scanner(System.in);
+
+ //If there is input, put input into string
+ input = in.nextLine();
+ in.close();
+
+ //Runs through the generation once without ouput to count the total number of clauses
+ int clauses = 0;
+
+ //Iterates over 3 loops so that it goes through every row in the sudoku board giving a number 1-9
+ for(int i=1; i < 10; i++) {
+ for(int n=1; n < 10; n++) {
+ clauses++;
+ }
+ }
+
+ //Iterates over 3 loops so that it goes through every column in the sudoku board giving a number 1-9
+ for(int j=1; j < 10; j++) {
+ for(int n=1; n < 10; n++) {
+ clauses++;
+ }
+ }
+
+ //Iterates over 5 loops so that it goes through every 3x3 grid in the sudoku board giving a number 1-9
+ for(int r=0; r < 3; r++) {
+ for(int s=0; s < 3; s++) {
+ for(int n=1; n < 10; n++) {
+ clauses++;
+ }
+ }
+ }
+
+ //Iterates over 4 loops so that it goes through every cell in the sudoku board making sure no cell has more than 1 number
+ for(int i=1; i < 10; i++) {
+ for(int j=1; j < 10; j++) {
+ for(int n=1; n < 9; n++) {
+ for(int m=n+1; m < 10; m++) {
+ clauses++;
+ }
+ }
+ }
+ }
+
+ //Adds extra input from STDIN if input is not empty
+ if(input.length() == 81) {
+ int stdinCount = 0;
+ while(stdinCount < input.length()) {
+ for(int r=1; r < 10; r++) {
+ for(int c = 1; c < 10; c++) {
+ //check to make sure character isn't '.'
+ if(input.charAt(stdinCount) != '.') {
+ clauses++;
+ }
+ stdinCount++;
+ }
+ }
+ }
+ }
+
+ //Print statement to generate the comment section.
+ System.out.print(
+ "c Daniel Shchur\n" +
+ "c\n" +
+ "c Sudoku CNF Problem\n" +
+ "c\n" +
+ "c Provides a Sudoku board to be solved\n" +
+ "c\n");
+
+ //Prints header
+ System.out.printf("p cnf %d %d\n", 9*9*9, clauses);
+
+ //Iterates over 3 loops so that it goes through every row in the sudoku board giving a number 1-9
+ System.out.println("c (1) Every row contains every number:");
+ for(int i=1; i < 10; i++) {
+ for(int n=1; n < 10; n++) {
+ for(int j=1; j < 10; j++) {
+ //Prints the combination as a single variable
+ System.out.printf("%d%d%d ", i, j, n);
+ }
+ //Prints the terminating character
+ System.out.println("0");
+ }
+ }
+
+ //Iterates over 3 loops so that it goes through every column in the sudoku board giving a number 1-9
+ System.out.println("c (2) Every column contains every number:");
+ for(int j=1; j < 10; j++) {
+ for(int n=1; n < 10; n++) {
+ for(int i=1; i < 10; i++) {
+ //Prints the combination as a single variable
+ System.out.printf("%d%d%d ", i, j, n);
+ }
+ //Prints the terminating character
+ System.out.println("0");
+ }
+ }
+
+ //Iterates over 5 loops so that it goes through every 3x3 grid in the sudoku board giving a number 1-9
+ System.out.println("c (3) Each 3x3 block contains every number:");
+ for(int r=0; r < 3; r++) {
+ for(int s=0; s < 3; s++) {
+ for(int n=1; n < 10; n++) {
+ for(int i=1; i < 4; i++) {
+ for(int j=1; j < 4; j++) {
+ //Prints the combination as a single variable
+ System.out.printf("%d%d%d ", (3*r + i), (3*s + j), n);
+ }
+ }
+ //Prints the terminating character
+ System.out.println("0");
+ }
+ }
+ }
+
+ //Iterates over 4 loops so that it goes through every cell in the sudoku board making sure no cell has more than 1 number
+ System.out.println("c (4) No cell contains more than one number:");
+ for(int i=1; i < 10; i++) {
+ for(int j=1; j < 10; j++) {
+ for(int n=1; n < 9; n++) {
+ for(int m=n+1; m < 10; m++) {
+ //Prints the combination as a single variable with terminating character
+ System.out.printf("%d%d%d %d%d%d %d\n", -i, j, n, -i, j, m, 0);
+ }
+ }
+ }
+ }
+
+ //Adds extra input from STDIN if string is not empty
+ if(input.length() == 81) {
+ int stdinCount = 0;
+ System.out.println("c (5) clauses generated from input");
+ while(stdinCount < input.length()) {
+ for(int r=1; r < 10; r++) {
+ for(int c = 1; c < 10; c++) {
+ //check to make sure character isn't '.'
+ if(input.charAt(stdinCount) != '.') {
+ System.out.printf("%d%d%s 0\n", r, c, input.charAt(stdinCount));
+ }
+ stdinCount++;
+ }
+ }
+ }
+ }
+
+
+ //End of code
+ }
+}
\ No newline at end of file
diff --git a/hwSat06/unsat.cnf b/hwSat06/unsat.cnf
new file mode 100644
index 0000000000000000000000000000000000000000..397141ec7b52226226bfbec9425be1a1edb25275
--- /dev/null
+++ b/hwSat06/unsat.cnf
@@ -0,0 +1,3197 @@
+c Daniel Shchur
+c
+c Sudoku CNF Problem
+c
+c Provides a Sudoku board to be solved
+c
+p cnf 729 3185
+c (1) Every row contains every number:
+111 121 131 141 151 161 171 181 191 0
+112 122 132 142 152 162 172 182 192 0
+113 123 133 143 153 163 173 183 193 0
+114 124 134 144 154 164 174 184 194 0
+115 125 135 145 155 165 175 185 195 0
+116 126 136 146 156 166 176 186 196 0
+117 127 137 147 157 167 177 187 197 0
+118 128 138 148 158 168 178 188 198 0
+119 129 139 149 159 169 179 189 199 0
+211 221 231 241 251 261 271 281 291 0
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+215 225 235 245 255 265 275 285 295 0
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+218 228 238 248 258 268 278 288 298 0
+219 229 239 249 259 269 279 289 299 0
+311 321 331 341 351 361 371 381 391 0
+312 322 332 342 352 362 372 382 392 0
+313 323 333 343 353 363 373 383 393 0
+314 324 334 344 354 364 374 384 394 0
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+911 921 931 941 951 961 971 981 991 0
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+918 928 938 948 958 968 978 988 998 0
+919 929 939 949 959 969 979 989 999 0
+c (2) Every column contains every number:
+111 211 311 411 511 611 711 811 911 0
+112 212 312 412 512 612 712 812 912 0
+113 213 313 413 513 613 713 813 913 0
+114 214 314 414 514 614 714 814 914 0
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+198 298 398 498 598 698 798 898 998 0
+199 299 399 499 599 699 799 899 999 0
+c (3) Each 3x3 block contains every number:
+111 121 131 211 221 231 311 321 331 0
+112 122 132 212 222 232 312 322 332 0
+113 123 133 213 223 233 313 323 333 0
+114 124 134 214 224 234 314 324 334 0
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+741 751 761 841 851 861 941 951 961 0
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+778 788 798 878 888 898 978 988 998 0
+779 789 799 879 889 899 979 989 999 0
+c (4) No cell contains more than one number:
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diff --git a/hwSat06/unsat.txt b/hwSat06/unsat.txt
new file mode 100644
index 0000000000000000000000000000000000000000..832ea0065b5b715286cdb83cf80cca4a033221ce
--- /dev/null
+++ b/hwSat06/unsat.txt
@@ -0,0 +1 @@
+..1..7.9.59..8...1.3.....8......58...5..6..2...41......8.....3.1.9.2..79.2.7..4..
\ No newline at end of file
diff --git a/hwSat06/unsatResults.txt b/hwSat06/unsatResults.txt
new file mode 100644
index 0000000000000000000000000000000000000000..abb3248efb654890006fa920d31b28cdb5585d38
--- /dev/null
+++ b/hwSat06/unsatResults.txt
@@ -0,0 +1 @@
+UNSAT