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\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 12 April 2018}
\maketitle

Complete the following problems. Fully justify each response.


\begin{enumerate}

\item Let $X$ be a set, and let $\sim_1$ and $\sim_2$ be two equivalence relations on $X$. Define a relation $\sim$ by $x\sim y$ if and only if $x\sim_1 y \wedge x\sim_2 y$. Prove that this is an equivalence relation. Describe the equivalence classes $[x]_\sim$ in terms of $[x]_{\sim_1}$ and $[x]_{\sim_2}$.

\item Let $P$ be a poset with element set $\N$, and order $\preceq$, where $a\preceq b$ if and only if $a|b$. Formally prove that this is a poset.

\item Let $P$ be a poset with partial order $\leq$. We define an element $m$ of $P$ to be {\it minimal} if there does not exist any $x\in P$ with $x<m$.
\begin{enumerate}
\item How is a minimal element different from a minimum element (as defined in class)?
\item Give an example of a poset that has at least one minimal element, but no minimum element.
\item Prove that if a poset has a minimum element, then it has only one minimal element.
\end{enumerate}
(Note: This entire problem could be repeated verbatim for a maximal element as well)

\item Let $X$ be a set, and let $P$ be the poset whose elements are $\mathcal{P}(X)$, and $A\leq B$ if and only if $A\subseteq B$.

Let $\{A_1, A_2, \dots, A_n\}$ be elements of $P$. Prove that $\displaystyle \bigwedge_{i=1}^n A_i = \displaystyle\bigcap_{i=1}^n A_i$, and $\displaystyle \bigvee_{i=1}^n A_i = \displaystyle\bigcup_{i=1}^n A_i$.

\item Let $(X, \preceq)$ be a lattice, and let $x, y\in X$. Prove the following:
\begin{enumerate}
\item $x\vee y = y \vee x$
\item $x\wedge y = y\wedge x$
\item $x\vee (x\wedge y)=x$
\item $x\wedge(x\vee y)=x$
\end{enumerate}

\item Give an example of a lattice that is not distributive.

\end{enumerate}

\end{document}