我正在用 LaTeX 撰写我最新的作业。
不幸的是，我的讲师给我的副本在 LaTeX 中无法“开箱即用”。
原来的：http://pastebin.tlhiv.org/7m9aiLWK 我的版本：http://pastebin.tlhiv.org/qgLkJU7T
因此，我尝试让它工作，但是却遇到了这个错误：
! Paragraph ended before \document was complete.<to be read again>\parl.69
非常感谢 marcellus 帮助我实现这个功能（FreeNode 上的#latex）：
\documentclass[a4paper,11pt]{article}
\usepackage[english]{babel}
\usepackage[pdftex]{graphicx}
\usepackage[all]{xy}
\usepackage{amsmath,amssymb,ntheorem}
\usepackage{multicol}
\title{}
\begin{document}
\begin{enumerate}
\item Negate the sentence $$\exists x\, \left(\forall y\, \left(Qx \wedge \left(\left(-Px \vee Qx\right)
\rightarrow Qy\right)\right)\right)\,.$$
Your final answer may involve $-Px$, $-Qx$, $-Py$ and $-Qy$, but should not use $-$ in any other context.
%%%
\item Write down the negation of the statement
$$
\exists x \in \mathbb{Z},\left(\exists y \in \mathbb{Z}, \left(\left(\left(x+1\right)^2 > y^2\right)
\rightarrow \left(x \ne y\right)\right)\right)
$$
Which, if any, of the two statements is true?
%%%
\item Suppose the universal set $\mathcal{E} = \{x \in \mathbb{N} \mid\ x \le 12\}$. Let $A = \{x\ \mid\ x \text{\ is\ odd}\}$,
$B = \{x\ \mid\ x > 7\}$ and $C = \{x\ \mid\ x$ is\ divisible by $3\}$.\newline
Write the each of following sets in enumerated form, then depict each using a Venn diagram.
\begin{multicols}{2}
\begin{enumerate}
\item $A \cap B$
\item $B \cup C$
\item $\overline{A}$
\item $(A \cup \overline{B}) \cap C$
\item $\overline{A \cup C} \cup \overline{C}$
\end{enumerate}
\end{multicols}{2}
%%%
\item Show that $\overline{\overline{A} \cap B} = A \cup \overline{B}$, using the laws of sets.
%%%
\eject
\item In a class of 35 students, there are 12 students who can speak German and 5 students who can speak Japanese. If 2 students can speak both
of these languages, how many students in the class can speak neither language? Justify your answer by constructing some appropriate sets and
considering their cardinality.
%%%
\item With the aid of a Venn diagram, find an expression for the cardinality of $A \cup B \cup C$, for any sets $A,B,C$.
\item Show that $G=(V,E)$ has no Hamiltonian cycle,
where $V=\{\,a,b,c,d,e,f,g\,\}$ and
$E=\{\,ab,ac,bc,bd,cd,de,df,ef,eg,fg\,\}$.
\item Prove that $G=(V,E)$ has no Eulerian walk,
where
\begin{align}
V&=\{\,a,b,c,d,e,f\,\} \text{ and}\\
E&=\{\,ab,ac,bc,bd,ce,de,df,ef\,\}.
\end{align}
Show that it is possible to add one edge to $G$ to form a graph that does have an Eulerian walk, and find such a walk.
\item %\bookmark%
Here is a weighted graph. The numbers in the table give
the weight of the edge joining each pair of vertices. Use Prim's
algorithm to find a minimal spanning tree in this weighted graph.
Then use Kruskal's algorithm to do the same thing.
% $$
% \paste{pics/assign2tbls}1
% $$
\end{enumerate}
The %\paste%
command above includes the output of the following code to typeset the table, produced with yet another tex variant.
% \boxedtables
% \sides
% \table
% \tss \g(10pt)\cmath\Black\g(10pt)\rt7{&\g(10pt)\cmath\g(10pt)}\\ \hl
% |b|c|d|e|f|g|h\\ \hl
% a|18|4|37|93|28|46|55\\ \hl
% b|\g(-10pt)\Gray\vrule width28pt\g(-10pt)\Black|47|12|9|51|72|63\\ \hl
% c|\xx2\g(-10pt)\Gray\vrule width57pt\g(-10pt)\Black|65|82|11|44|33\\ \hl
% d|\xx3\g(-10pt)\Gray\vrule width85pt\g(-10pt)\Black|94|17|61|56\\ \hl
% e|\xx4\g(-10pt)\Gray\vrule width114pt\g(-10pt)\Black|29|73|38\\ \hl
% f|\xx5\g(-10pt)\Gray\vrule width142pt\g(-10pt)\Black|31|5\\ \hl
% g|\xx6\g(-10pt)\Gray\vrule width171pt\g(-10pt)\Black|59\\ \hl
% \endtable
\end{document}