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\begin{document}
\title{Math 301 Homework}
\author{Mary Radcliffe}
\date{Due 15 February 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $p$ and $q$ be propositional variables. Prove that $p\iff q$ is logically equivalent to $\neg\left( ((\neg p)\wedge q)\vee (p\wedge (\neg q))  \right)$.

\item Suppose it is known that every continuous function on a closed interval $[a,b]$ in $\R$ has a maximum value. Now, suppose I have a function $f$, which I tell you has a maximum value on the interval $[a,b]$. What can you conclude about the continuity of $f$? Use the language of propositional logic to explain your answer.

\item Let $p(x, y)$ be the statement that $xy=1$. Suppose that $x$ is a member of the positive integers $\{1, 2, 3, \dots\}$ and $y$ is a member of the rationals. Consider the following two statements:
\begin{enumerate}
\item $\forall x, \exists y, p(x,y)$
\item $\exists y, \forall x, p(x, y)$
\end{enumerate}
How do these statements differ? Are either of them true?

\item For each of the following statements about sets, determine if the statement is true or false. If true, prove the statement. If false, explain why.
\begin{enumerate}
\item For any set $X$, $\emptyset\in X$.
\item For any set $X$, $\emptyset\subseteq X$.
\item $\{x\in \Z\ | \ x\geq 0\} = \N$.
\item $\emptyset \subseteq \mathcal{P}(\emptyset)$.
\item $\emptyset \in \mathcal{P}(\emptyset)$.
\end{enumerate}

\item Prove that $X\subseteq Y$ if and only if $X\cap Y = X$.

\item Prove that $X\cup(Y\cap Z) = (X\cup Y)\cap (X\cup Z)$.
\end{enumerate}

\end{document}