Hykilpikonna
188 lines
6.4 KiB
TeX
188 lines
6.4 KiB
TeX
\documentclass[fontsize=11pt]{article}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage[utf8]{inputenc}
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\usepackage[margin=0.75in]{geometry}
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\title{CSC110 Fall 2021 Assignment 2: Logic, Constraints, and Nested Data}
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\author{Azalea Gui & Peter Lin}
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\date{\today}
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\begin{document}
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\maketitle
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\section*{Part 1: Predicate Logic}
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\begin{enumerate}
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\item[1.]
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\begin{enumerate}
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\item[1.] When $D_1 = [0,\infty) $ \\
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Statement 1 is True because every number $x \in D_1$ is smaller than a $y \in D_1$ (For example, $y=x+1>x$). \\
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Statement 2 is False because when $y=0$, there isn't an $x \in D_1$ smaller than $y$.
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\item[2.] When $D_2 = \mathbb{Z}$ \\
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Statement 1 is True because every integer $x$ is smaller than some integer $y$ (For example, $y=x+1>x$). \\
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Statement 2 is True because every integer $y$ is greater than some integer $x$ (For example, $x=y-1<y$).
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\item[3.] When $D_3 = \{0\}$ \\
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Statement 1 is False because when $x=0$, there isn't a $y \in D_3$ greater than $x$. \\
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Statement 2 is False because when $y=0$, there isn't an $x \in D_3$ smaller than $y$.
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\end{enumerate}
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\item[2.]
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\begin{enumerate}
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\item[1.] $P(x): 1 < x < 8$ where $x \in S$
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\item[2.] $Q(x): x < 8$ where $x \in S$
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\end{enumerate}
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When $x \leq 1$ and $x \geq 8$, $P(x)$ is False, so $P(x) \land Q(x)$ would be False and $P(x) \implies Q(x)$ would be True. \\
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When $1 < x < 8$, $P(x)$ and $Q(x)$ are both True, so both $P(x) \land Q(x)$ and $P(x) \implies Q(x)$ would be True. \\
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Since $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, \\
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Statement 3 would be False because $P(x) \land Q(x)$ is False when $x=0$, \\
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Statement 4 would be True because $P(x) \implies Q(x)$ is true for all values of $x \in S$.
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\item[3.]
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Complete this part in the provided \texttt{a2\_part1.py} starter file.
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Do \textbf{not} include your solution in this file.
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\item[4.]
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Complete this part in the provided \texttt{a2\_part1.py} starter file.
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Do \textbf{not} include your solution in this file.
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\end{enumerate}
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\section*{Part 2: Conditional Execution}
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Complete this part in the provided \texttt{a2\_part2.py} starter file.
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Do \textbf{not} include your solution in this file.
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\newpage
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\section*{Part 3: Generating a Timetable}
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\begin{enumerate}
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\item[1.]
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Complete this part in the provided \texttt{a2\_part3.py} starter file.
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Do \textbf{not} include your solution in this file.
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\item[2.]
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\begin{enumerate}
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\item[(a)]
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\emph{IMPORTANT DEFINITIONS/NOTATION} (don't change this text!)
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We define the following sets:
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\begin{itemize}
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\item $C$: the set of all possible courses
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\item $S$: the set of all possible sections
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\item $M$: the set of all possible meeting times
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\item $SC$: the set of all possible schedules
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\end{itemize}
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We also define the following notation for expressions involving the elements of these sets:
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\begin{itemize}
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\item
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The first three (courses/sections/meeting times) are represented as tuples (as described in the assignment handout), and you can use the indexing operation on these values. For example, you could translate ``every section term is in $\{'F', 'S', 'Y'\}$'' into predicate logic as the statement:
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\[\forall s \in S,~ s[1] \in \{'F', 'S', 'Y' \} \]
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\item
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The start and end times of a meeting time can be compared chronologically using the standard $<$, $\leq$, $>$, and $\geq$ operators.
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\item
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For a section $s \in S$, $s[2]$ represents a tuple of meeting times.
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You may use standard set operations and quantifiers for these tuples (pretend they are sets).
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For example, we can say:
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\begin{itemize}
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\item $\forall s \in S,~ s[2] \subseteq M$
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\item $\forall s \in S,~ \forall m \in s[2],~ m[1] < m[2]$
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\end{itemize}
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\item
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Finally, for a schedule $sc \in SC$, you can use the notation $sc.sections$ to refer to a set of all sections in that schedule.
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You can use quantifiers with that set of schedules as well, e.g.
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$\forall s \in sc.sections,~ ...$
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\end{itemize}
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\textbf{Predicate for meeting times conflicting:}
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% TODO: fill in the predicate definition for two meeting times conflicting
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\begin{align*}
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MeetingTimesConflict(m_1, m_2) : m_1[0] == m_2[0] \land m_1[2] > m_2[1] \land m_2[2] > m_1[1] \\
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\qquad \text{where $m_1, m_2 \in M$}
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\end{align*}
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\smallskip
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\textbf{Predicate for sections conflicting:}
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% TODO: fill in the predicate definition for two sections conflicting.
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% Use the MeetingTimesConflict predicate in your response.
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\begin{align*}
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SectionsConflict(s_1, s_2) : (s_1[1]=\text{'Y'} \lor s_2[1]=\text{'Y'} \lor s_1[1] = s_2[1]) \land \\
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\exists m_1 \in s_1[2], \exists m_2 \in s_2[2], \text { s.t. } MeetingTimesConflict(m_1, m_2) \\
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\qquad \text{where $s_1, s_2 \in S$}
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\end{align*}
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\smallskip
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\textbf{Predicate for valid schedule:}
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% TODO: fill in the predicate definition for a schedule being valid.
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% Use the SectionsConflict predicate in your response.
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\begin{align*}
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IsValidSchedule(sc) : \forall s_1, s_2 \in sc.sections, SectionsConflict(s_1, s_2) \Rightarrow s_1 = s_2
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\qquad \text{where $sc \in SC$}
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\end{align*}
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\item[(b)]
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Complete this part in the provided \texttt{a2\_part3.py} starter file.
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Do \textbf{not} include your solution in this file.
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\end{enumerate}
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\item[3.]
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\begin{enumerate}
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\item[(a)]
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You may use all notation from question 2(a).
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Note that a course $c \in C$ is a tuple, and $c[2]$ is a set of sections, and so can be quantified over: $\forall s \in c[2], ...$.
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\smallskip
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\textbf{Predicate for section-schedule compatibility:}
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% TODO: fill in the predicate definition for a section being compatible with a schedule.
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\begin{align*}
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IsCompatibleSection(sc, s) : \forall s_1 \in sc.sections, \neg SectionsConflict(s, s_1)
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\qquad \text{where $sc \in SC, s \in S$}
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\end{align*}
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\smallskip
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\textbf{Predicate for course-schedule compatibility:}
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% TODO: fill in the predicate definition for a course being compatible with a schedule.
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% Use IsCompatibleSection in your response.
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\begin{align*}
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IsCompatibleCourse(sc, c) : \exists s \in c[2] \text{ s.t. } IsCompatibleSection(sc, s)
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\qquad \text{where $sc \in SC, c \in C$}
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\end{align*}
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\item[(b)]
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Complete this part in the provided \texttt{a2\_part3.py} starter file.
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Do \textbf{not} include your solution in this file.
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\end{enumerate}
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\end{enumerate}
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\section*{Part 4: Processing Raw Data}
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Complete this part in the provided \texttt{a2\_part4.py} starter file.
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Do \textbf{not} include your solution in this file.
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\end{document}
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