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\title[Math 250 Handout 14  - equivalence relations]{Math 250 Handout
  14  - equivalence relations}\maketitle


Which of the following are equivalence relations? (Which are
  reflexive, symmetric, or transitive?)
\begin{enumerate}



\item  Let $x$ and $y$ be real numbers and define $x \sim y$ if $x -
  y \in \mathbf{Q}$.


\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}

\item Let $x$ and $y$ be rational numbers and define $x \sim y$ if $x -
  y \in \mathbf{Q}_{\geq 0}$.


\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}

\item   Let $x$ and $y$ be integers and define $x \sim y$ if $x -
  y \in d\mathbf{Z}$. 

\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}
  
  \item Let $S$ be the collection of all sets and say that $A \sim
  B$ if there is a bijection from $A$ to $B$.

\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}

\item Let $S$ be the collection of all sets and say that $A \sim
  B$ if there is a surjection from $A$ to $B$.

\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}

\item Let $S$ be the collection of all sets and say that $A \sim
  B$ if there is an injection from $A$ to $B$.


% \vspace{.4cm}
%   \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
% \vspace{.9cm}

% \item Let $S$ be the collection of all sets and say that $A \sim
%   B$ if $A \cap B$ is empty.


\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}



\item Let $x$ and $y$ be real numbers and define $x \sim y$ if $x
  = 1$ or $y = 1$.


\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}

\item Let $x$ and $y$ be real numbers and define $x \sim y$ if $x
  = 1$ or $y = -1$.


\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}

\item Let $\mathbf{Q}[x]$ be the set of polynomials with rational
  coefficients. Say that $f \sim g$ if their derivatives are equal.


\vspace{.4cm}
  \hspace{.4cm} R \hspace{.4cm} S \hspace{.4cm} T
\vspace{.9cm}
\end{enumerate}
\end{document}




