\documentclass[minion]{homework}

% Some common macros
\newcommand{\Reals}{\ensuremath{\mathbb R}}% Gives you a shortcut for writing the blackboard R for the real numbers - \RR
\newcommand{\Nats}{\ensuremath{\mathbb N}} % Gives you a shortcut for writing the blackboard N for the natural numbers - \NN
\newcommand{\Ints}{\ensuremath{\mathbb Z}} % Gives you a shortcut for writing the blackboard Z for the integer numbers - \ZZ

\doclabel{Math F404: Homework 1}
\docdate{Due: September 2, 2026}

\begin{document}
\begin{problems}
\problem Sutherland 2.1

\problem Sutherland 2.2

\problem Sutherland 2.5 \textbf{(AI-free)}

\problem Sutherland 2.7 as follows:

Suppose $\sim$ is an equivalence relation on a set $X$.
Show that the equivalence classes partition $X$ in the 
following sense.  Let $\mathcal C$ denote the collection of
equivalence classes.  Then $\cup_{C\in\mathcal C} C =X$,
and if $C,C'\in\mathcal C$ then either $C=C'$ or $C\cap C'=\emptyset$.

\problem Give examples of collections of open intervals $I_i=(a_i,b_i)$
with $i\in \Nats$ such that 
\begin{subproblems}
\item $\cap_{i\in\Nats} I_i = [0,1]$
\item $\cap_{i\in\Nats} I_i = (0,1)$
\item $\cap_{i\in\Nats} I_i = (0,1]$
\end{subproblems}
For part (a) give a formal proof.  For the remainder, just give an example and a brief justification.

\problem Sutherland 3.2 (No formal proof needed)

\problem Sutherland 3.3

\problem Sutherland 3.4

\problem Sutherland 3.8 (Make your proof as short as possible)

\end{problems}

\end{document}
