\documentclass{amsart}
\usepackage{amssymb, amsmath}
\begin{document}
\begin{center}
{\Huge 18.100C Homework 2}
\\
Due online by noon on Monday, September 28
\end{center}
\medskip
This assignment has four parts. All four parts should be submitted in .tex and .pdf formats through the Stellar course website. Parts 1-3 should be in a single document and Part 4 in a separate document.
\begin{enumerate}
\item For each of the following statements, write down the inverse, converse and contrapositive. Indicate which of the statements are logically equivalent to each other due to a simple rule of logic. State (but you need not prove) which statements are true and which are false.
\begin{enumerate}
\item If $r + s$ is rational then $r$ is rational or $s$ is irrational.
\item If $r$ is irrational or $s$ is irrational then $r + s$ is irrational.
\item If $A$ is open then $A^C$ is closed. (Take as given that $A$ is in the universe of subsets of a given metric space $X$.)
\item If $0 < x < y$ then $0 < \dfrac{1}{y} < \dfrac{1}{x}$. (Take as given that $x$ and $y$ are elements of a given ordered field $\mathbb{F}$.)
\end{enumerate}
\item Write down the negation of each of the following statements.
\begin{enumerate}
\item $\forall \varepsilon > 0, x \in A, \, \exists y \in B$ such that $d(x, y) < \varepsilon$. (Take as given that $A$ and $B$ are subsets of some metric space $(X, d)$.)
\item $\forall \varepsilon > 0 \, \exists \delta > 0\, \forall f \in F, x \in X, y \in X \, \Big((d_1(x, y) < \delta) \implies$ \\ $(d_2(f(x), f(y)) < \varepsilon)\Big)$. (Take as given that $F$ is a set of functions whose domain is the metric space $(X, d_1)$ and whose range is another metric space $(Y, d_2)$.)
\end{enumerate}
\item Construct the truth table for the statement $(P \wedge Q) \implies (P \vee Q)$.
\item TeX up one of the solutions (your choice) to your regular 100B/C problem set from the assignment due Friday, September 25. (This will be part of every future recitation homework assignment.)
\end{enumerate}
\end{document}