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%\title{\textbf{4113: Math Econ}}%
%\author{Instructor: Sasha Vostroknutov}%
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\section{Preference Relations}
Suppose we have a binary relation $R$ (you can use any other symbol, for example $\succsim$) on the set of alternatives $X$. Remember that $R\subseteq X^2$ and for $x,y\in X$ we say that $xRy$ is true if $(x,y)\in R$. Here are some definitions. You should read the statements below as follows: $R$ is \textbf{reflexive} if for all $x\in X$ it is true that $xRx$.

\begin{itemize}
\item[] \textbf{reflexive} if $xRx$
\item[] \textbf{irreflexive} if $\neg (xRx)$
\item[] \textbf{transitive} if $xRy\ and\ yRz\Rightarrow xRz$
\item[] \textbf{total} if $x\neq y\Rightarrow [xRy\ or\ yRx]$
\item[] \textbf{complete} if $xRy\ or\ yRx\ or\ both$
\item[] \textbf{symmetric} if $xRy\Rightarrow yRx$
\item[] \textbf{asymmetric} if $xRy\Rightarrow \neg yRx$
\end{itemize}

To understand these definitions think of two preference relations $"="$ and $">"$ defined on the real line. Consider $=$ first: it is reflexive ($x=x$ is true for any $x\in\Real$); transitive ($x=y=z$ implies $x=z$); symmetric ($x=y$ implies $y=x$); not total ($x\neq y$ does not imply $x=y$ or $y=x$). Now think of $>$: it is irreflexive ($\neg (x > x)$ is true); transitive ($x>y>z$ implies $x>z$); total ($x\neq y$ implies either $x>y$ or $y>x$); asymmetric ($x>y$ implies not $y>x$).

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