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\title{Homework \#6}
\author{Date: March 18, 2020 : Due: April 1, 2020 }
\begin{document}
\maketitle

In class we have been talking about relations and we have defined
the the notions of reflexive, symmetric and transitive.
A relation $R$ on a set $D$ is called {\it anti-reflexive} if $(x,x) \notin R$ for
any $x \in D$.
A relation $R$ is called {\it anti-symmetric} if $(x,y) \in R$ with $x \neq y$,
then $(y,x) \notin R$.  A relation is called
{\it circular} if $(x,y) \in R$ and $(y,z) \in R$, then $(z,x) \in R$.

\begin{enumerate}
\item Prove that it is not possible to have a relation on a non-empty set $D$ which is both reflexive and anti-reflexive.
Note that if $D$ is empty, it is possible to have a relation which is both reflexive and anti-reflexive
so make sure that your explanation reflects this.\\
\item Choose $D = \{1,2,3,4\}$ to be the domain of a relation.  For each of the 8 combinations 
\begin{align*}
\{ (A, B, C) : &A \in \{ \hbox{anti-reflexive}, \hbox{not anti-reflexive} \},\\
&B \in \{ \hbox{anti-symmetric}, \hbox{not anti-symmetric} \},\\
&C \in \{ \hbox{circular}, \hbox{not circular} \} \}
\end{align*}
give at least one example of a relation which satisfies exactly those possibilities.
For some combinations of properties, it may be
that no such relation exists.  In these cases,
prove that it is impossible by providing an explanation why.\\
\item A {\it partially ordered set} (sometimes called a {\it poset}) is a set $D$ together with a relation on $D$ which is
reflexive, anti-symmetric and transitive.  Prove that the positive integers together with the relation 
$$\{ (a,b) : a,b \in {\mathbb Z}, a\hbox{ divides }b \}$$
is a partially ordered set.\\
\item Prove that the relation on the complex numbers defined by
$$R = \{ (x,y) : x,y \in {\mathbb C}, x = a y\hbox{ for some }a \in {\mathbb R}, a>0\}$$
is an equivalence relation.  Give a description of the equivalence classes of the
relation.
\end{enumerate}
\end{document}