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\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
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\newtheorem{ex}{Exercise}
\newtheorem*{defn}{Definition}

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\begin{document}

\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 8\\
\today
\end{flushright}

%Please write your solutions in the designated solution area!

\begin{ex}
Let $A$ and $B$ be sets inside some universal set. Prove DeMorgan's Law
$$(A\cap B)^c=A^c\cup B^c.$$
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $A_1,A_2,...,A_n$ be a collection of subsets of some universal set. Prove the extended version of DeMorgan's Law
$$(A_1\cap A_2\cap \cdots\cap A_n)^c = A_1^c\cup A_2^c\cup\cdots\cup A_n^c.$$
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $A$ and $B$ be sets. Prove:
\begin{center}
If $A\subset B$, then $A\cap B^c=\varnothing$.
\end{center}
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove for sets $A,B,C$
$$A\times(B\cap C) = (A\times B)\cap(A\times C).$$
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove or disprove: For sets $A,B,C$
$$(A\cap B)\cup C = A\cap(B\cup C).$$
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Fill in the blanks with the properties (from the class handout) used to prove this result
$$\begin{array}{rclrr}
(A\cup B)\setminus (C\setminus A) & = & (A\cup B)\cap(C\setminus A)^c  & \text{by} & \underline{(a)}\\
                                  & = & (A\cup B)\cap(C\cap A^c)^c     & \text{by} & \underline{(b)}\\
                                  & = & (A\cup B)\cap(A^c\cap C)^c     & \text{by} & \underline{(c)}\\
                                  & = & (A\cup B)\cap((A^c)^c\cup C^c) & \text{by} & \underline{(d)}\\
                                  & = & (A\cup B)\cap(A\cup C^c)       & \text{by} & \underline{(e)}\\
                                  & = & A\cup(B\cap C^c)               & \text{by} & \underline{(f)}\\
                                  & = & A\cup(B\setminus C)            & \text{by} & \underline{(g)}\\
\end{array}$$
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Give an ``algebraic proof'' (as in the previous exercise) of the identity
$$((A^c\cup B^c)\setminus A)^c = A.$$
\end{ex}

\begin{proof}

\end{proof}

\begin{defn}[Topology]
Let $X$ be a set. A \emph{topology} on $X$ is a collection, $\mathcal{T}$, of subsets of $X$ such that
\begin{enumerate}
\item $\varnothing,X\in\mathcal{T}$
\item For any subcollection $\{V_1,V_2,V_3,...\}$ of $\mathcal{T}$
$$\bigcup_{i=1}^\infty V_i\in\mathcal{T}.$$
\item For any \emph{finite} subcollection $\{V_1,V_2,...,V_n\}$ of $\mathcal{T}$
$$\bigcap_{i=1}^n V_i\in\mathcal{T}.$$
\end{enumerate}
\end{defn}

\begin{ex}
Let $X$ be a set. Show that $\PS(X)$ satisfied the definition to be a topology on the set $X$. (This is called the \emph{discrete topology on $X$}.)
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $X=\{a,b,c,d,e\}$ and $Y=\{1,2,3,4,5,6\}$. Define the function $f:X\to Y$ by
$$f=\{(a,3),(b,5),(c,2),(d,3),(e,6)\}.$$
\begin{enumerate}[$(a)$]
\item What is the domain and co-domain of $f$?
\item Give the values of $f(a)$, $f(b)$, and $f(e)$.
\item What is the range of $f$?
\item Is $c$ the inverse image of $2$?
\item What is the inverse image of $3$? the inverse image of $4$?
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $f:X\to Y$ be a function. Let $A$ and $B$ be subsets of $X$. Below are two statements, one of which is true and one of which is false. Decide which is true and which is false. Give a counterexample to verify the false one, and prove the true one.
\begin{eqnarray*}
f(A\cap B)&=&f(A)\cap f(B)\\
f(A\cup B)&=&f(A)\cup f(B)
\end{eqnarray*}
\end{ex}

\begin{proof}

\end{proof}

\end{document}