In order for this relation to be an equivalence equation, it has to be reflexive, symmetric and transitive.

\textbf{Reflexive:}

All elements are related to themself

\[ AA, DD, CC, EE, BB \]

\textbf{Symmetric:}

All relations has its symmetric counterpart

\begin{gather*}
AE\text{ and }EA \\
AC\text{ and }CA \\
BD\text{ and }DB \\
CE\text{ and }EC
\end{gather*}

\textbf{Transitive:}

All pair of relations where $xRy$ and $yRz$ has its transitive counterpart

\begin{gather*}
DB\text{ and }BD\text{ with }DD \\
AE\text{ and }EA\text{ with }AA \\
AE\text{ and }EC\text{ with }AC \\
AC\text{ and }CE\text{ with }AE \\
AC\text{ and }CA\text{ with }AA \\
BD\text{ and }DB\text{ with }BB \\
CE\text{ and }EA\text{ with }CA \\
CE\text{ and }EC\text{ with }CC \\
EA\text{ and }AE\text{ with }EE \\
EA\text{ and }AC\text{ with }EC \\
EC\text{ and }CE\text{ with }EE \\
EC\text{ and }CA\text{ with }EA \\
CA\text{ and }AE\text{ with }CE \\
CA\text{ and }AC\text{ with }CC
\end{gather*}

Hence the relation is an equivalence relation