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\centerline{\textbf{Honors HW04. Morally DUE Mon Mar 14}}

A linear ordering $L$ has the {\bf Levin property} if the following hold:
\begin{itemize}
\item 
There exists a $MIN$ element.

Formally

$$(\exists x)(\forall y)[x \le y].$$

In later problems we will call this $x$ $MIN$.

\item
There exists a $MAX$ element.

Formally

$$(\exists y)(\forall x)[x \le y].$$

In later problems we will call this $x$ $MAX$.

\item
For all $y\ne MIN$ there is an element $x$ such that $x<y$ and there is nothing inbetween $x$ and $y$. 

Formally

$$(\forall  y\ne MIN)(\exists x)[ x<y \wedge (\forall z)[(z\le x) \vee (z\ge y)]].$$


\item
For all $x\ne MAX$ there is an element $y$ such that $x<y$ and there is nothing inbetween $x$ and $y$. 

Formally

$$(\forall  x\ne MAX)(\exists y)[ x<y \wedge (\forall z)[(z\le x) \vee (z\ge y)]].$$
\end{itemize} 

\begin{enumerate}
\item 
(50 points) 
Give an example of an ordering $L$ with the Levin Property such that E wins the
Emptier-Filler game with ordering $L$.
\item 
(50 points) 
Give an example of an ordering $L$ with the Levin Property such that F wins the
Emptier-Filler game with ordering $L$.
\end{enumerate}

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